Extending two-Higgs-doublet models by a singlet scalar field — The case for dark matter

We extend the two-Higgs doublet models of Type I and Type II by adding a real gauge-singlet scalar S dark matter candidate (2HDMS models). We impose theoretical constraints deriving from perturbativity, stability, unitarity and correct electroweak symmetry breaking and require that the lightest CP-even Higgs, h, fit the LHC data for the ∼ 125.5 GeV state at the 68% C.L. after including existing constraints from LEP and B physics and LHC limits on the heavier Higgs bosons. We find that these models are easily consistent with the LUX and SuperCDMS limits on dark-matter-Nucleon scattering and the observed Ωh2 for S masses above about 55 GeV. At lower mS, the situation is more delicate. For points with mS in the 6-25 GeV range corresponding to the CDMS II and CRESST-II positive signal ranges, the dark-matter-Nucleon cross sections predicted by the Type I and Type II models more or less automatically fall within the 95%–99% C.L. signal region boundaries. Were it not for the LUX and SuperCDMS limits, which exclude all (almost all) such points in the case of Type I (Type II), this would be a success for the 2HDMS models. In fact, in the case of Type II there are a few points with 5.5 GeV ≲ mS ≲ 6.2 GeV that survive the LUX and SuperCDMS limits and fall within the CDMS II 99% C.L. signal region. Possibilities for dark matter to be isospin-violating in this 2HDMS context are also examined.


Introduction
One of the most important extensions of the Standard Model (SM) is the inclusion of additional particle(s) that comprise the dark matter (DM) of the Universe. A particularly important possibility is a weakly-interacting-massive-particle (WIMP) with thermal relic density consistent with current observations. An important constraint on the WIMP scenario are limits on the spin-independent WIMP-nucleon cross section, σ SI , the strongest of which are currently those of the LUX [1] and SuperCDMS [2] Collaborations, where the LUX limit is strongest for DM masses above about 6 GeV while the SuperCDMS limit is JHEP11(2014)105 strongest for masses below this. 1 In combination, the LUX and SuperCDMS limits exclude the positive CDMS II signal observed for a WIMP with mass of ∼8. 6 GeV and cross-section of σ SI ∼ 1.9×10 −41 cm 2 [4,5] as well as the other positive hints (DAMA [6], CoGeNT [7,8], and CRESST-II [9]) that support the findings of CDMS II. We note that isospin-violating DM (IVDM) scenarios [10] that could make the Xenon-based LUX limit consistent with the CDMS II Silicon-based positive signal [11][12][13][14] do not appear to be relevant given that the SuperCDMS Germanium-based limits require only minor rescaling [15,16].
In this paper we focus on a one-component DM model in which the WIMP is a singlet scalar particle that is present as part of an extended scalar sector of the electroweak theory. In particular, we consider two-Higgs-doublet models (2HDM) with an extra real scalar S (we term the resulting models "2HDMS") that is neutral under the SM gauge group. 2 We introduce an extra Z 2 symmetry under which S is the only odd field. Provided S does not acquire a vacuum expectation value (VEV), it is stable and thereby a possible DM candidate. The 2HDMS then contains three CP-even states, h and H (m h ≤ m H ) from the 2HDM sector and S, a CP-odd state, A, and a charged Higgs pair, H ± . The 2HDM context allows for increased flexibility for DM predictions as compared to adding an S to the one-doublet SM in that either h or H can be identified with the observed SM-like CPeven state at ∼ 125.5 GeV while the other CP-even state and the A and H ± can provide additional channels for early-universe annihilation. Further, both of the CP-even states contribute to DM scattering and annihilation.
That the 2HDM can provide a consistent description of all LHC observed signal strengths for either the h or H identified as the observed ∼ 125.5 GeV state (for the Type I or Type II version of the model) is well-known [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. For simplicity, in this paper we consider only the case of m h ∼ 125.5 GeV. In the context of DM, the crucial new ingredient offered by 2HDMS is the presence of two independent Higgs portal couplings, H † 1 H 1 SS and H † 2 H 2 SS, where H 1,2 are the two Higgs doublets of the 2HDM. As will be discussed in detail later, this is an important feature that makes it possible to decouple DM annihilation from DM scattering off nucleons. It also provides more freedom while trying to overcome constraints from invisible decays of the 125.5 GeV Higgs boson in a multiple scalar singlet extension of the SM [35] or in the two component DM scenario of [36]. The singlet extension of the 2HDM has been studied earlier in [37][38][39][40][41][42][43][44][45] and also discussed in the frameworks of scale invariance [46] and non-SUSY SO(10) grand unification (GUT) [47][48][49]. The primary new ingredient in the present paper is the inclusion of the full set of constraints on the 2HDM sector of the 2HDMS. These include: requiring consistency with "preLHC" constraints; an accurate fit to the combined ATLAS and CMS Higgs signal data when the h is identified with the ∼ 125.5 GeV state; and enforcing LHC limits on the other Higgs bosons (H, A and H ± ) of the 2HDM using the procedures of [32]. With regard to the singlet sector, we derive and employ the constraints on the singlet parameters resulting 1 We note that the XENON 100 limit [3] is weaker than the LUX limit for all dark-matter masses and, thus, we do not reference it in our discussions. 2 Here we will restrict ourself to the CP-conserving version of the 2HDM. However, this assumption is just to reduce the number of parameters. The analysis could as well be performed assuming either spontaneous or explicit violation of CP in the scalar sector.

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from imposing perturbativity, stability, unitarity and correct electroweak symmetry breaking. As implicit from the H † 1 H 1 SS and H † 2 H 2 SS Higgs portal interactions, SS annihilation via both the h pole and the H pole will be accounted for.
In the following section, we will summarize the fits to preLHC and LHC data within the 2HDM context with m h ∼ 125.5 GeV. In section 3, we discuss the two-Higgs-doublets plus singlet model (2HDMS), including its general features and theoretical constraints as well as the properties of the singlet dark matter scalar. In section 4 we elaborate on the methodology of constraining the full 2HDMS parameter space using various experimental observations and limits when the 2HDM sector of the model is restricted to fit existing LHC data. In section 5 we will present the results of our 2HDMS parameter space scan. There, we show that the combined LUX and SuperCDMS DM limits can only be satisfied for m S > ∼ 55 GeV. However, we do explore the extent to which IVDM scenarios arise in the 2HDMS case and how they come close to allowing the CDMS II signal to be consistent with the LUX limit. Section 6 contains our conclusions. In appendices A and B we derive the constraints on the 2HDMS from vacuum stability and unitarity, respectively.

Fitting the 8 TeV LHC Higgs signal in the 2HDM
The combined ATLAS and CMS data imply that the observed ∼ 125.5 GeV state is quite consistent with SM-like Higgs boson. Recent 2HDM efforts [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] have thus focused on the extent to which deviations from the SM are still possible and the implications for possibly observing such deviations and/or the other Higgs bosons in future LHC running. Of course, one must keep in mind that there is still an enhanced γγ signal in the ATLAS analysis whereas γγ rates are somewhat suppressed according to the CMS analysis and it is only the combined results that show no γγ enhancement. Should an enhancement become statistically certain in future LHC runs, this could certainly be accommodated in the 2HDM context [50][51][52][53][54][55][56][57][58][59], as could a suppression, but the analysis performed in this paper would have to be revisited. In this paper, we take the combined data at face value and employ the very recent 2HDM fits of [32] keeping only points that are consistent with observations at the (rather stringent) 68% C.L., assuming that it is the lighter h that should be identified with the observed ∼ 125.5 GeV state. To be specific, the predicted signal strengths in the µ(ggF + ttH) versus µ(VBF + VH) planes for each of the γγ, V V (where V V ≡ ZZ, W W ), bb, and τ τ final states were required to have χ 2 < 2.3 as determined using the C.L. contours established in [60].
The parameters of 2HDM can be taken to be the mixing angle, α, that diagonalizes the CP-even scalar sector, 3 cos α/ sin β cos α/ sin β cos α/ sin β −sin α/ cos β H cos(β − α) sin α/ sin β sin α/ sin β sin α/ sin β cos α/ cos β A 0 cot β − cot β cot β tan β Table 1. Tree-level vector boson couplings C V (V = W, Z) and fermionic couplings C F (F = U, D) normalized to their SM values for the Type I and Type II two-Higgs-doublet models.  Figure 1. 2HDM points in the (tan β, sin α) plane that provide a fit the LHC/Tevatron signal strengths at 95% C.L. (cyan) and 68% C.L. (dark green), from the analysis in [60]. In red we have marked the 68% C.L. points used later in the singlet scalar model analysis (for the Type II model we have used all 68% C.L. points).
values, of the Higgs bosons to vector bosons (C V ) and to up-and down-type fermions (C U and C D ) are functions of α and β as given in table 1; see e.g. [62] for details. The Type I and Type II models are distinguished only by the pattern of their fermionic couplings.
When expanding the 2HDM to include an extra singlet that could be dark matter, it is appropriate to begin with 2HDM points that provide a good fit to the LHC data. As noted above, we assume that it is the lighter h that should be identified with the 125.5 GeV state and take the 2HDM points from [32] that provide a fit to the LHC data within 68% C.L. These points, along with the points agreeing at the less restrictive 95% C.L., are shown in figure 1 using the tan β vs. sin α plane. (Because there are so many 68% C.L. points in the Type I 2HDM we employ only a subset of these points in this case -the full 68% C.L. set of points are shown in dark green while the selected points are shown in red.) Of course, in order that the LHC fit for m h ∼ 125.5 GeV be good, the vector boson and fermionic couplings (see table 1) should be quite SM-like. The exact SM limit occurs for β −α = π/2. The extent to which 68% C.L. allows deviation in these couplings is illustrated in figure 2 where we plot the ratios of these couplings to their SM values, C h V for the V V coupling and C h D for the down-quark. (For Type I, C h U = C h D .) We observe that in the case of Type II almost all points have C h V and C h D (and C h U , not plotted) very close to unity (whereas at 95% C.L. significant deviations are allowed). In the case of Type I, significant deviations in these couplings from unity are still allowed at 68% C.L.
When adding in the singlet S we thus must be certain that it will not significantly disturb the fit of the h to the LHC data. Because of the extra imposed Z 2 symmetry, the only influence of the S on the h fits arises if h → SS decays are present, which of course requires m S < m h /2. These would constitute invisible decays. In [60] a 68% C.L. limit of BR inv ≤ 0.1 (see also [63,64]) was obtained in the context where the C U , C D and C V coupling ratios could be varied with respect to their SM values of unity (but with C V ≤ 1 as appropriate to a 2HDM) and assuming no extra loop contributions to the hγγ and hgg couplings. In the 2HDM, the H ± loops can contribute to the hγγ coupling, but for simplicity we will assume that BR inv ≤ 0.1 remains applicable. The constraint of small BR inv = BR(h → SS) plays a major role in eliminating many m S < m h /2 scenarios.

2HDMS models
Our goal is to analyse a model with two Higgs doublets H 1 , H 2 and a real scalar S, which is a singlet under the SM gauge group. We will assign equal U(1) Y charges Y = 1 to H 1 and H 2 . We also introduce a Z 2 symmetry under which S → −S (other fields are taken to be even under Z 2 ). We call this model 2HDMS. 4 The most general gauge-invariant 2HDMS scalar potential is then: which contains 20 (real) parameters. However, for simplicity we make several additional assumptions. We consider a model without explicit CP violation (i.e. all the λ coefficients of eq. (3.1) are taken to be real) and we only consider parameter choices for which there is no spontaneous CP breaking. As a result, the Higgs VEVs are real. We also impose a Z 2 symmetry under which H 1 → H 1 , H 2 → −H 2 , S → S. This eliminates the λ 6 , λ 7 and κ 3

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couplings of eq. (3.1). However, we do allow for m 2 12 = 0, corresponding to a soft breaking of Z 2 . The resulting potential takes the form The next stage is to convert from the Lagrangian basis to the mass eigenstate basis. Despite the presence of the S 2 H † 1 H 1 and S 2 H † 2 H 2 terms, the analysis of the 2HDM sector can be performed independently of the S and the usual mass matrices for the 2HDM, see [61], are not changed due to the fact that the extra field S does not acquire a VEV. 5 However, the fields H 1 and H 2 do contribute to the S 2 mass term when they develop VEVs, In terms of the mass eigenstates, the S-dependent part of the scalar potential has the form: where the physical S particle mass and the DM-Higgs trilinear couplings are λ h = −κ 1 sin α cos β + κ 2 cos α sin β (3.5) λ H = κ 1 cos α cos β + κ 2 sin α sin β . While m 0 , κ 1 and κ 2 constitute a complete set of extra (as compared to the 2HDM) free parameters for the scalar sector of the 2HDMS Lagrangian, in practice it is more convenient to employ the DM mass m S and the couplings λ h and λ H as the new independent set of free parameters associated with the S sector. In the limit of sin(β − α) = 1, for which the h has exactly SM-like couplings to V V and f f , We also emphasize that although there is no ASS term in V S due to CP, the CP-odd Higgs boson A still plays a role in determining the DM relic density through the creation/annihilation process SS ←→ AA. We will discuss this issue in section 4.

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The quadrilinear couplings λ HH , λ hH , λ hh , λ AA , λ H + H − can also be expressed in terms of the κ 1 , κ 2 , α and β parameters: We note that the above Lagrangian-level trilinear and quadrilinear couplings convert to Feynman rules according to: The fermionic couplings in the 2HDMS depend upon the behavior of the fermionic fields under Z 2 and Z 2 . We assume that the fermions are even under Z 2 so that the S has no tri-linear coupling to f f . 6 Fermionic couplings to H 1 and H 2 depend on the Z 2 signs for fermions. We choose these so as to forbid flavour-changing Yukawa couplings for the neutral Higgs bosons, resulting in the couplings of table 1 for the models of Type I and Type II. From now on, we restrict ourselves to the Z 2 × Z 2 case.
Further constraints on the model are as follows.

Perturbativity
All quartic Feynman rules associated with the mass eigenstates h, H, A, H ± , S are required to satisfy the standard perturbativity constraint, i.e. their absolute values must be ≤ 4π. As regards the sector involving the S field, the quartic couplings of interest are those in which S 2 multiplies two 2HDM fields and the S 4 term. One can show that the quartic Feynman rules connecting S 2 to two neutral 2HDM fields, summarized above, are guaranteed to be smaller than 4π in absolute value if |κ 1 |, |κ 2 | ≤ 4π is imposed. However, these maximum values are only allowed for α = ±π/4. The Feynman rule for S 4 interactions being λ S means that we must also impose 0 < λ S ≤ 4π, the lower bound being that required for stability.

Vacuum stability
We require that the vacuum is stable at tree level, which means that the potential in (3.2) has to be bounded from below. As already noted this requires first of all that λ S > 0. Given this, it is shown in appendix A that the necessary and sufficient conditions for stability read:

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If κ 1 or κ 2 < 0, then we have to satisfy also: The conditions in eq. (3.12) above are the standard 2HDM stability conditions. These are supplemented by the requirements of eq. (3.13), eq. (3.14) and eq. (3.15) in the presence of the singlet field.

S-matrix unitarity
In addition, there are constraints deriving from unitarity that are closely correlated with the constraints from perturbativity. Indeed, the dominant non-vanishing contributions to amplitudes for two-body scattering at high energy come from the processes mediated by quartic couplings. Therefore, the unitarity constraint for J = 0 partial waves, |a 0 | ≤ 1/2, reduces to a constraint on these quartic couplings. In appendix B, we describe in more detail the unitarity bounds and give explicit formulae for the scattering matrix of two-body processes in the scalar sector of the 2HDMS model. 7

Electroweak Symmetry Breaking (EWSB)
In order to ensure a stable DM particle S, one has to require S = 0 at the global minimum of the scalar potential, eq. (3.2). For each 2HDM point at 68% C.L. (marked in red in figure 1), tan β and m 2 12 are given and all five λ's can be computed from the masses of the Higgs bosons and sin α (see details in appendix D of [61]). With these specified, the remaining parameters m 1 and m 2 in the potential, eq. (3.2), are determined by the minimization conditions Note that the minimization with respect to S is trivial because of S = 0. In practice, we find all the minima of eq. (3.2) numerically and then eliminate the points for which the global minimum is not at S = 0, H 1 = 0, H 2 = 0. In figure 3, the allowed regions in the (κ 1 , κ 2 ) parameter space are displayed after sequentially imposing the various constraints discussed above. i) At the first level, we impose perturbativity (P). All subsequent point layers obey P. 7 It is important to note that the 2 → 2 scattering matrix that is obtained when S-related channels are included always has a maximum eigenvalue that is larger than that of the pure 2HDM 2 → 2 scattering matrix. This is called the "bordering theorem" (see e.g. [66]). Thus, although our 2HDM points have already been filtered using the 2HDMC code [67,68] which imposes unitarity in the 2HDM context, the unitarity limits obtained after including the S-related channels are guaranteed to be stronger.

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Figure 3. The plot shows the impact of the perturbativity (P), vacuum stability (S), unitarity (U) and electroweak symmetry breaking (EW) global minimum bounds discussed in section 3 on the (κ 1 , κ 2 ) plane. At the first level, the grey points are those which satisfy P -all subsequent point layers obey P. Note that |κ 1 |, |κ 2 | ≤ 4π contains the perturbative region (see section 3.1). Subsequent point layers were plotted in the following order: points after the stability bound, S (green), points after the unitarity bound, U (orange), points after the stability and unitarity bounds, S+U (black), points after the stability, unitarity and EW bounds, S+U+EW (red). The value of the λ S parameter was set to 4π (0.1) in the upper (lower) plots. In this figure, no restriction on m S is imposed.
ii) Next, we require vacuum stability (S).
S is always guaranteed as long as κ 1 and κ 2 are both positive. For κ 1 < 0 and/or κ 2 < 0, vacuum stability depends on the value of the S self-interaction coupling λ S .
Choosing the maximum value of λ S = 4π (upper panel), there is an ellipse-shaped region of modest size where κ 1 and/or κ 2 can be negative. This ellipse-shaped region shrinks as λ S decreases -we illustrate this for the case of λ S = 0.1 in the lower panel.
iii) Third, the unitarity conditions (U) on their own produce an oval-shaped region in the κ 2 vs. κ 1 plane. v) Imposing S+U+EW eliminates some of the (black) S+U points, leaving us with the red points. Figure 4 shows how the above κ 1 vs. κ 2 regions map into the λ h vs. λ H parameter space. In this figure, no restriction on m S is imposed. In fact, the P+S+U+EW constraints are much more restrictive for m S < m h /2. This is illustrated in figure 5. In particular, note that the maximum value of λ H that is allowed is of order 3 in magnitude, at large λ S , and is very tiny for small λ S . As a result, very large values of m H cannot result in sufficient annihilation through the H pole diagram when m S < m h /2 given that the h pole diagram is suppressed because λ h must be very small in order to avoid too large BR(h → SS).

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Of course, P+S+U+EW are only the most basic constraints. In the following sections, we will show that once Ωh 2 is required to agree with observations, then |λ h | and |λ H | are restricted to values < ∼ 0.2 and < ∼ 2.5, respectively. When m S < m h /2, BR(h → SS) ≤ 0.1 further constrains |λ h | to values < ∼ 0.01.

Experimental constraints on 2HDMS
Before starting our analysis of the model, we would like to summarize the experiments that impact the extra singlet S particle.

Dark matter relic abundance
In the 2HDMS, the S particle provides the only candidate for DM and thus should comprise the total relic abundance of the early Universe. To a good approximation, the relic density is given by where x f = m S /T f 20 is the typical freeze-out temperature of a WIMP [69], M P l is the Planck mass, g * is number of relativistic degrees of freedom, σv is the thermally averaged cross section for SS annihilation into the SM particles (i.e. leptons and quarks , ff , and gauge bosons, W + W − , ZZ, denoted collectively as XX) and into Higgs bosons (hh, hH, HH, AA, H + H − ). The Feynman diagrams for all the processes are shown in JHEP11(2014)105 figure 6. First, the process of annihilation into the SM particles is mediated by an schannel h or H only. Following [38] (see also [43], which however has small numerical factor errors), we find where C H X is the coupling of H to XX relative to the coupling of the SM Higgs boson to XX and Γ SM (H * → XX) stands for the SM partial width in the XX final state calculated at invariant mass √ s = 2m S . (Note: for X = Z, then X = Z also. In this case, Γ(H * → XX) must include the 1/2! for identical particles in the final state.) In this equation, the total width, Γ H , must include the width for H → SS and any partial width modifications relative to the SM width for the various SM channels (in particular, the enhancement of Γ(H → bb) at large tan β in the Type II case.) Second, there are all the channels containing Higgs pairs. For the (H i H i ) = (AA) or (H + H − ) final states, the relevant diagrams are the first two diagrams in the upper row of the figure, which include not only s-channel h or H exchange but also a four-point contact self-coupling. For final states containing CP-even Higgs pairs, (H i H j ) = (hh), (HH), (hH), there are contributions from t-and u-channel S exchange (the last two diagrams with different topologies in the top row of figure 6) in addition to the s-channel h or H exchange diagrams and the four-point contact self-coupling. A formula that applies to all these different cases is most easily given in terms of the Feynman rules for the various relevant vertices: the quartic Feynman rules were given earlier in eq. (3.11) and the trilinear coupling g HhH Feynman rule can be found in appendix F of [61]. We find Note that some final states will typically be kinematically closed. In particular, for m S < m h only the f f (f = t), V V and, possibly, AA channels will be allowed. In order to illustrate results of the scan over singlet parameter space, in figures 7 and 8 we show Ωh 2 as a function of m S for representative 2HDM points when scanning over the remaining singlet parameters. The 2HDM parameters for these four points are given in table 2. For the first case, figure 7, the 2HDM parameters are such that low m S is eliminated JHEP11(2014)105  when correct EWSB is imposed in addition to stability and unitarity. In the second case,  . Results for the relic abundance Ωh 2 as a function of m S coming from a scan over the singlet parameter space for a fixed 2HDM point. The sample 2HDM parameters employed are given in table 2. All points satisfy perturbativity as defined earlier. Black points satisfy the stability and unitarity conditions, red points satisfy also the EWSB conditions. Blue points satisfy S+U+EW and have BR(h → SS) ≤ 0.1. The yellow band is the recent ±3σ Planck window Ωh 2 = 0.1187 ± 0.0017 at 68% CL [70]. We emphasize that the LUX and other limits on DM detection are not yet imposed in these plots.

Higgs invisible/unseen decays
In addition to decays into SM particles, the CP-even Higgs bosons h and H of the 2HDMS have a number of possible invisible and/or "unseen" decays. By "unseen" we mean decay modes that contain visible particles, but that the experimental analyses have not explored and/or are not yet able to place useful limits on. The invisible decays are h, H → SS and the potentially important unseen decay modes are h → AA and H → AA, hh. Since we assume that it is the h that is the ∼ 125.5 GeV state, we are not immediately concerned with H decays. However, both h → SS and h → AA decays could make it impossible to fit the LHC Higgs data at the 68% C.L. level that we are requiring. In fact, at this level of fitting precision, the scans of [32] did not find points with m A < m h /2. Thus, we JHEP11(2014)105 need only ensure that, for each point in the full 2HDMS parameter space, BR(h → SS) is sufficiently small as to not significantly disturb the fit of the h to the LHC Higgs data. The h, H → SS decay widths are given by: where i = 1, 2 denotes h, H and the dimensional Feynman-rule couplings g h i SS are given in eq. (3.11). In what follows, it will be most convenient to discuss results in the space of the dimensionless λ H vs. λ h parameters, where g h i SS = −2λ h i v. When the decay h → SS is kinematically open, it will dominate the decay of the h unless λ h is very small. Large BR(h → SS) would invalidate the fits to the LHC 125.5 GeV signal. The constraints on such an invisible decay are thus quite strong: BR(h → SS) ≤ 10% at 68% C.L. [60]. In practice, this bound is violated for most m S < 55 GeV points in the full 2HDMS parameter space leaving only a small number of points with λ h 1 for which BR(h → SS) ≤ 0.1. This is illustrated in figure 9, which shows points in the (λ h , λ H ) plane, coloured with respect to the resulting BR(h → SS). Invisible decays of the H will be discussed later.

Direct detection
The rate at which DM-particles scattering off nuclei can be detected is directly related to the DM-nuclei scattering cross-section [71], which is given by:  Table 3. Form factors extracted from micrOMEGAs 3.0. where q is the momentum transfer, µ r = (m N m S )/(m N +m S ) and v is the relative velocity. The couplings of DM to the proton and neutron, f p and f n , can be expressed as where we have used g h i qq = −i gmq 2m W C h i q (m W = 1 2 gv in our convention) with the quark coupling factors C h i q for Type I and II models as listed in table 1 and the Feynman rule g h i SS expressions given in eq. (3.11). In practice, direct detection rates in our calculation have been evaluated using micrOMEGAs [72], including QCD NLO corrections.
There are numerous collaborations (LUX, XENON 100, SuperCDMS, CDMS, CoGeNT and DAMA being of particular interest to us) working on the direct detection of DM. They typically translate the limit on the event rate against recoil energy they directly detect into a limit on the DM-proton cross section σ DM−p as a function of DM mass. However, in reality there are several standard assumptions hidden in this translation that might or might not be correct. For instance, they assume a DM halo in the vicinity of Earth and employ the truncated Maxwell-Boltzmann velocity distribution below the escape velocity obtained from the Standard Halo Model. They also assume that the DM particle elastically scatters with a short range contact interaction via a 'heavy mediator', implying zero-momentum JHEP11(2014)105 transfer. Of particular importance, they adopt the assumption that DM has equal coupling to the neutron and proton, that is to say the ratio f n /f p = 1 .
Indeed, this equality approximately holds in the Type I model because of the universal coupling structure with up-type and down-type quarks, see table 1. In fact, λ SSqq mq is independent of quark-species and the common couplings C h U,D and C H U,D in the Type I model can be factored out and will then cancel out in the ratio. From eqs. (4.8) and (4.9), one can then derive the ratio of f n /f p in the Type I case: However, the relation f n /f p = 1 is not always true in the Type II model. In order to compare the predicted cross-sections for DM-nucleon scattering with the results presented by the experimental groups, we define the nucleon-normalized cross section, σ DM−p , following [10,12]: where σ DM−p is the predicted DM-proton cross-section and the rescaling factor Θ X is defined as single isotope detector , multiple isotope detector (4.12) where I runs over all isotopes present in the detector X and η I is the relative abundance of the I'th isotope. Note that if f n /f p = 1, then Θ X (f n , f p ) = 1. However, when f n /f p = 1, Θ X (f n , f p ) will depend upon the isotope abundances and is therefore determined by the properties of the chemical elements used in the various detectors. It was pointed out in [11][12][13][14] that the scattering amplitudes of DM with proton and neutron may interfere destructively in such a way as to achieve f n /f p ∼ −0.7, the value for which the resulting LUX exclusion limits are not in strong conflict with the favored signal regions of the Silicon-based CDMS II experiment and the Germanium-based CoGeNT experiment. However, these positive signal regions are in direct conflict with the limits obtained by SuperCDMS [15,16]. In any case, in order to interpret any given DM scattering result, it is necessary to compute f n /f p for each Type II parameter point. Further, f n /f p in general depends on the singlet sector parameters. However, there is an interesting special case in which f n /f p depends only on the 2HDM parameters. Recalling that the positive CDMS II and CoGeNT signals are both at rather low m S ∼ 6 − 12 GeV and noting that BR(h → SS) will be large for such masses unless JHEP11(2014)105 Figure 11. f n /f p in the limit of λ h = 0 as a function of sin α and tan β for the 68% C.L. Type II 2HDM scan points. λ h is very small, it is useful to give an approximation for f n /f p in the limit of λ h → 0, i.e. in the limit of ignoring the h term in eq. (4.8). In this limit, the value of f N depends only on the quark couplings of the H: In figure 11, we display the resulting f n /f p as a function of sin α for the Type II points from [32] that give Higgs boson property fits at the 95% C.L. or better. There, we see a large range of f n /f p values, ranging from +1.5 to ∼ −0.9. However, for the 68% C.L. Type II points that we include in our study, points with substantially negative f n /f p are rather sparse, with the most negative value associated with a single isolated point close to −0.7. This is just an accidental result given the scanning procedure/density employed in [32].
The rather singular structure of this plot can be understood as follows. Since the LHC data at 68% C.L. are in good agreement with SM predictions, most of the Type II 2HDM points shown in figure 1 have β − α π/2, in which case C H u − cot β, C H d tan β in the Type II model. In this approximation, one can use eq. (4.13) to obtain tan β as a function of f n /f p in the limit of λ h → 0: For the value f n /f p = −0.7, one finds tan β = 1.04364 implying α − π 4 and sin α ∼ −0.707, with a small variation associated with the exact form factor values. Although we have a single point with these approximate values, it turns out that for BR(h → SS) ≤ 0.1

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the possible Ωh 2 values lie outside the 3σ window that we have allowed. Thus, within the limitations of the scanning so far performed we have not managed to produce a point that satisfies all the constraints not related to DM-scattering that also has f n /f p ∼ −0.7, but we regard it as possible that much denser scans might reveal a point of this type. Of course, to the extent that we accept the SuperCDMS upper bound, the CDMS II result is excluded in any case given that all the acceptable points have f n /f p values that are close to 1. So, it is perhaps a good feature of the 2HDMS model that obtaining a point consistent with f n /f p ∼ −0.7 and all other constraints requires a very fine-tuned choice of tan β and sin α.

DM full mass scan
As noted earlier, instead of scanning over the full 2HDMS parameter space, for simplicity we used selected points from the 2HDM phenomenologically allowed points of [32] (labelled as "postLHC8-FDOK"), as outlined in section 2. In the case of m h ∼ 125.5 GeV, the 2HDM analysis of [32] found ∼ 5200 points consistent with Higgs observations at 68% C.L. in the Type I model, from which we randomly chose 1250 points for further analysis. For the Type II model we use all of the ∼ 900 points that fall within the 68% C.L. criterion. These points are marked in red in figure 1. For each surviving 2HDM point, we perform a scan over the extra singlet parameters: m S , λ h , λ H . We then check theoretical constraints for the 2HDMS model including perturbativity, stability, unitarity and proper electroweak symmetry breaking, as discussed in section 3. Since the extra scalar S does not acquire a VEV, it does not mix with the other Higgs bosons h and H. As a result, the experimental constraints from electroweak precision tests (STU parameters), B physics, direct searches at LEP and also limits on the heavier Higgs bosons (H and possibly A) are barely influenced by the presence of the singlet scalar S. Therefore, the postLHC8-FDOK points in the 2HDM can be adopted as good starting points when expanding to the 2HDMS. As we have noted, the only caveat that arises is the need to take into account the possibility of h → SS decays when the scalar S is light. Substantial BR(h → SS) will spoil the pure 2HDM fit performed in [32]. Including limits from the current Higgs invisible decay searches at the LHC one finds roughly that BR(h → SS) ≤ 30%(10%) is required at 95%(68%) C.L. Therefore, as discussed earlier, we impose a cut of BR(h → SS) ≤ 10% for all points presented in the following context (except for a few situations as described later) in order to maintain the LHC signal fit and consistency with invisible decay limits. Finally, we use micrOMEGAs [72] to calculate the relic abundance of the DM candidate S and require that the predicted Ωh 2 fall within the ±3σ Planck window Ω exp DM h 2 = 0.1187 ± 0.0017 at 68% C.L. [70]. Hereafter, we refer to this set of constraints as the "preLUX" constraints.
Let us now turn to the issue of DM scattering on nuclei. For the points satisfying the "preLUX" constraints, we calculate the cross section for the scattering of the S off a nucleon and compare the predicted value σ DM−p (after rescaling by Θ in the case of Type II) to the latest LUX limits for the DM-proton cross section, denoted σ LUX DM−p (which are obtained assuming f n /f p = 1). If the points obey the condition σ DM−p ≤ σ LUX DM−p , they are not excluded by the LUX limit. Recall that for Type I, f n /f p ∼ 1 and so no rescaling is required between target types. Also shown are contours corresponding to the CRESST-II, CoGeNT and CDMS II positive signal regions. In the case of CRESST-II, the darker black contour is at 68% C.L. and the lighter grey contours are at 95% C.L. In the case of CoGeNT (orange region) we show only the 90% C.L. contour. For CDMS II, we display contours (using various levels of grey) at 68%, 90%, 95% and 99% C.L.

Type I analysis
In figure 12 we present the cross section versus m S for the Type I model. Since f n /f p ∼ 1 in the case of the Type I model, all experimental results can be displayed on the same plot. Points obeying the LUX limit are shown in green. Points that do not pass the LUX limit but do satisfy all preLUX conditions (including correct Ωh 2 and BR(h → SS) ≤ 0.1) are shown in blue. Note that few green points at very low m S that pass the LUX limit are excluded by the SuperCDMS limit. Note that the Type I predictions for σ DM−p agree pretty well with CDMS II/CRESST-II data (for more detailed discussion, see section 5.3.1, but, of course, disobey the LUX limit. The narrowness of the σ DM−p band at low m S can be understood as follows. In this mass region, we know that λ h 0, DM annihilation and scattering off nucleons are thus realized via H exchange in the s-and t-channels, respectively. Both processes are essentially controlled by the ratio λ H /m 2 H . We observe that once the constraints of BR(h → SS) ≤ 10% and good Ωh 2 are both satisfied λ h and λ H are roughly fixed. As a result, the predicted value of σ S−p as a function of m S is constrained to a narrow band that happens to pass through the CDMS II/CRESST-II preferred regions. However, the CDMS II/CRESST-II regions are simply not consistent with the combination of LUX and SuperCDMS limits in the Type I model. Finally, once m S > ∼ 55 GeV essentially all of the points that are consistent with preLUX constraints also pass the LUX limit (SuperCDMS limits do not extend to masses > ∼ 40 GeV).
In figure 13, we display the associated λ h and λ H values as a function of m S . We see that for m S < ∼ 50 GeV, the restriction to small λ h coming from requiring BR(h → SS) ≤ 0.1 implies that SS → H → SM will be dominant and correct Ωh 2 then requires relatively substantial λ H , the precise value depending on m H , see eq. (4.2). In contrast, there is a considerable variety of possibilities for λ h and λ H in the "resonance" region, i.e. in the vicinity of m S ∼ m h /2. Typically, both the h and H s-channel diagrams contribute to SS → XX. Once m S is above the resonance region, many channels open up and λ h is no longer restricted by a limit on BR(h → SS). A significant range of λ h becomes possible, the larger the value of m S the larger the range. Note that only in the resonance region are large values of λ H possible. There, contributions from h and H exchange can partially cancel. The fact that neither λ h nor λ H can be very large above the resonance region reflects the large number of final states that become available, in particular the hh channel opens up once m S > ∼ m h .

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We note that the "band" structure in the λ H vs. m S plot in the m S < ∼ 50 GeV region is due to the fact that H exchange is dominant for SS → XX annihilation. One finds that each band is associated with a particular m H value for the associated 2HDM point. As expected from eq. (4.2), the larger the value of m H the larger the value of λ H that is needed for correct Ωh 2 .

Type II analysis
We now turn to the Type II model. A particularly interesting question is whether or not one can have consistency between the CDMS II/CRESST-II preferred regions and the LUX limits. As already noted, this requires f n /f p ∼ −0.7. As a first step, we examine the correlation between the ratio of f n /f p and BR(h → SS), as illustrated in figure 14. After imposing the constraint BR(h → SS) ≤ 10%, as well as all the other preLUX constraints, all points with f n /f p ∼ −0.7 in the low m S region are excluded. Indeed, in the low m S region f n /f p > ∼ 1. Even relaxing the invisible decay limit to BR(h → SS) ≤ 55% (the most conservative upper bound on BR inv at the LHC [73]) still does not allow for points with f n /f p ∼ −0.7. In the resonance region of m S 55 GeV, a predicted f n /f p values range from below −1 to above 2, although the majority of points have f n /f p near 1. Above the resonance region, i.e. m S > ∼ m h /2, most points have f n /f p ∼ 1, but there is a handful of points with f n /f p values both substantially above 1 and substantially below 1 (a few JHEP11(2014)105 points have quite negataive values). Thus, in our predictions for DM scattering, it will be important to take into account the variation of f n /f p .
In order to present the overall picture for Type II, we adopt the parameters in [10] to calculate the rescaling factor Θ Xe for the Xenon-based detectors and present the σ DM −p cross sections in figure 15. In the left plot, we impose all preLUX constraints (including Ωh 2 in the 3σ window) other than BR(h → SS) ≤ 0.1. Points with f n /f p ∼ 1 (for which Θ Xe ∼ 1) are singled out as are points with f n /f p ∼ −0.7. Comparing with the right plot, one can find that only the former points can have BR(h → SS) ≤ 0.1, and only a subset of these can obey the LUX limits. Basically, we find that obtaining correct Ωh 2 while at the same time having BR(h → SS) ≤ 0.1 (or even ≤ 0.55) is not possible for the f n /f p ∼ −0.7 points in the low-m S region.
To explore in more detail the level of inconsistency between the LUX and SuperCDMS limits and the positive signal regions for CDMS II and CoGeNT, we present figure 16 which focuses on the m S ≤ 35 GeV mass range. All plotted points obey the full set of preLUX constraints (including BR(h → SS) ≤ 0.1). For the left figure, we have rescaled the DM-proton scattering cross section predicted for a given point by the factor Θ X , see eq. (4.12), as computed for X = Si in order to compare to the positive signal region found by the CDMS II Silicon detector. We also display the relevant limits from the SuperCDMS experiment. These are f n /f p dependent. The two lines correspond to the SuperCDMS limit after rescaling from the SuperCDMS Germanium target to the CDMS-II Silicon target. We rescaled σ SuperCDMS Si = σ SuperCDMS Θ Si (f n , f p )/Θ Ge (f n , f p ) using f n /f p = 1.05 and 1.25the minimum and maximum values shown in figure 14 for m S ≤ 35 GeV when BR(h → SS) ≤ 0.1 is imposed. We see that for the predicted range of f n /f p the resulting rescaling JHEP11(2014)105 Figure 15. Cross section for DM -proton scattering for the Type II model rescaled by the function Θ X defined in eq. (4.12), where X = Xe for a Xenon-based detector. All points plotted satisfy the preLUX constraints except BR(h → SS) ≤ 0.1 (i.e. they satisfy the theoretical constraints for 2HDMS, 2HDM fitting at 68% C.L. and the constraint on Ωh 2 ). In the left-hand plot, for the light purple points the ratio f n /f p is within the range (0.95, 1.05). For the darker purple points −0.8 ≤ f n /f p ≤ −0.6. The right-hand plot displays points that obey BR(h → SS) ≤ 0.1 in blue (i.e. they obey the full set of preLUX constraints), while the orange points obey only the weaker limit of BR(h → SS) ≤ 0.55. Figure 16. Cross section for DM-proton scattering for the Type II model rescaled by the function Θ X defined in eq. (4.12), where X = Si for a Silicon detector (CDMS II) on the left and X=Ge for the Germanium detector (CoGeNT) on the right. All points satisfy all the preLUX constraints (i.e. they satisfy the theoretical constraints for 2HDMS, 2HDM fitting at 68% C.L., BR(h → SS) ≤ 0.1 and the constraint on Ωh 2 ). The CDMS II contours shown are at 68%, 90%, 95% and 99% C.L. The CoGeNT contour is the 90% C.L. level contour. Light green points are allowed by LUX results. The larger black points are those allowed by both SuperCDMS and LUX and that also lie within the 99% C.L. CDMS II contour. The pink and light pink lines (almost degenerate) correspond to the SuperCDMS limit, after rescaling from the SuperCDMS Germanium target to the CDMS-II Silicon target using f n /f p = 1.05 and 1.25 (the minimum and maximum values shown in figure 14 for BR(h → SS) ≤ 0.1 when m S ≤ 35 GeV). Also shown by the dark green lines is the rescaled LUX limit, σ LUX Si = σ LUX Θ Si (f n , f p )/Θ Xe (f n , f p ), using the same two f n /f p values.

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is fairly minimal and those two limits are almost degenerate. Also shown by another two lines is the rescaled LUX limit, σ LUX Si = σ LUX Θ Si (f n , f p )/Θ Xe (f n , f p ), using the same two f n /f p values. From this plot, we observe that there are a few points (the large black points) with m S ∼ 5.5 − 6.2 GeV that lie below both the rescaled LUX limits and rescaled SuperCDMS limits. Further, although these points lie below the 2σ (95% C.L.) contour of the positive signal region of CDMS II, they do fall within the 3σ (99% C.L.) contour. Thus, the 2HDMS Type II model allows consistency between the CDMS II signal region (at 99% C.L.) and the SuperCDMS and LUX limits for a small range of low m S .
It is perhaps important to understand the points in figure 16 with low m S that obey LUX and SuperCDMS constraints in the case of the Type II model. Their properties appear in table 4. All have low tan β, very modest m H with m A , m H ± somewhat larger (in the 300 − 600 GeV range). It is worth recalling that for each 2HDM phenomenologically allowed point, the 2HDM parameters including tan β, sin α, m h , m H , m A , m H ± and m 12 are fixed. We then randomly scan over the singlet sector parameters κ 1 , κ 2 (or equivalently λ h , λ H ) and m S . Therefore, one can have many values of m S and corresponding rescaled cross section (the pair of numbers appearing in the last column of table IV) for each fixed 2HDM point whose parameters are listed in the first 6 columns.
For the right figure, we rescale σ DM−p using Θ X as computed for X=Ge in order to compare to the potential signal region for the CoGeNT Germanium detector. We find points consistent with all pre-LUX constraints within the CoGeNT 90% C.L. signal region for m S ∼ 10 − 15 GeV. However, the entire CoGeNT signal region is excluded by the SuperCDMS limit (no relative rescaling required since both are for a Germanium target) and by the LUX limit as indicated by the point coloring (where these limits have been rescaled using the f n /f p value for a given point to determine whether or not the point is excluded).
In the case of both the CDMS II figure and the CoGeNT figure, we note that allowing BR(h → SS) larger than 0.1 does not allow points much above those already shown, but rather increases the density of points where points are already shown.
As in the case of Type I, we could plot λ h and λ H vs. m S for the Type II points that obey preLUX constraints. The resulting point distributions look very similar to those shown in figure 13.

Summaries
It is perhaps useful to summarize what Type I and II models predict with regard to the invisible decays of the heavier H and how this will impact possibilities for detecting the H in upcoming LHC runs. For m S < ∼ 55 GeV, the BR(h → SS) ≤ 0.1 constraint required by a good h fit to the 125.5 GeV data implies that λ h is small and this indirectly impacts BR(H → SS). Before imposing the LUX limits, we find that BR(H → SS) can have a number of semi-discrete values below 1, the discreteness being associated with particular 2HDM 68% C.L. points, but for the bulk of m S < ∼ 55 GeV points one has BR(H → SS) > ∼ 0.9. Of course, we have seen above that once the LUX and SuperCDMS limits are imposed all the low-m S points are eliminated in the Type I case, whereas in the Type II case a handful of points survive in the m S ≤ 6 GeV region. Once m S > ∼ 55 GeV,  As regards H detection, we first note that since the HV V couplings are small (since the hV V coupling must be large for a good Higgs fit) the Z + inv final state LHC data do not currently constrain BR(H → SS), and in future runs very high integrated luminosity would be needed to have any hope of seeing a signal in this channel. Further, if H → SS decays are dominant this would reduce the strength of the H signals in other production/decay modes, such as gg → H → τ τ , and thus decrease the prospects for H discovery as outlined in [32]. In such instances, experimental sensitivity to the H may have to rely on gg → H production with a jet or photon tag of the invisible H → SS final state.
We now turn to an expanded discussion of the summary given above in which we split the scalar mass m S into three regions, depending on the status of the exotic decay h → SS: • low mass region (1 − 55 GeV) where the decay is open and could be substantial without λ h being very small; • resonance region (55 − 70 GeV) where m S is not far from the h pole location. For m S < m h /2, one finds that, after imposing P+S+U+EW, λ h is sufficiently limited that BR(h → SS) ≤ 0.1. In fact, in this region, the strongest constraint on λ h comes from the need to avoid too much annihilation.
• high mass region (70 − 1000 GeV) where the decay is absolutely closed.
Note that we adopt different scan strategies in these regions of m S so as to achieve a maximum density around the most interesting points that pass all theoretical and experimental constraints. The scans are also preformed in a different way for Type I and Type II models.

Low mass region
As we have already noted, in the low mass region, the exotic decay h → SS could have a large branching ratio. In the case where a singlet scalar is added to the pure SM, one finds that the corresponding coupling of dark matter to the Higgs necessary to avoid overabundance of the relic S is so large that BR(h SM → SS) > ∼ 0.9 [35,40], thereby making a good fit of the h SM to the LHC Higgs data impossible. In the 2HDMS, one can keep BR(h → SS) small enough (≤ 0.1) to avoid destroying the fit of the h to the 125.5 GeV Higgs data if λ h 1. Nonetheless, correct Ωh 2 can be achieved because in the 2HDMS the annihilation of DM is mediated not only by h but also by H (see figure 6). Therefore, the desired large cross section for SS annihilation can be achieved if λ H is sufficiently large when λ h is small. This trend was already apparent in figure 9. Here, we zero in on the m h ≤ 55 GeV region in figure 17, where we have employed a special scan strategy designed to cover a large range of f n /f p and small λ h . In the upper plots in figure 17, we require m S ≤ 50 GeV while the lower plots are for 50 < m S ≤ 55 GeV. In the latter case, we observe a hole in the vicinity of small λ h and λ H which expands to a gap in the former case due to the fact that points with m S ≤ 50 GeV are sufficiently far from the resonance region that H exchange, i.e. λ H = 0, is needed for correct Ωh 2 . In contrast, for points with JHEP11(2014)105 50 < m S ≤ 55 GeV, for λ h = 0 the h alone can provide enough annihilation for correct Ωh 2 even if λ H = 0.
As expected, the temperature plots show that, generally speaking, the larger m H is the larger λ H must be for correct relic density (the SS annihilation amplitude containing the ratio λ H /m 2 H ). However, there is an exception in the case of the Type II model; at large tan β ( > ∼ 25) one can have sufficient annihilation even if λ H /m 2 H is not large since the Hbb coupling is highly enhanced, C H D ∝ tan β, see eq. (4.2). We observe a smattering of such points in the (upper) m S ≤ 50 GeV Type II plot. For these points, the SS → bb annihilation cross section is large enough to produce relic abundance within the experimental limit even though |λ H | < 0.2 and m H > 500 GeV.
We end this subsection with the plots of figure 18 showing the regions of the 2HDM parameter space with m S ≤ 55 GeV that remain after imposing the full set of preLUX constraints. The allowed regions are displayed in the (tan β, sin α), (m H , m A ) and (m H ± , m A ) planes. Different colors are used to distinguish those points with m S ≤ 50 from those with 50 < m S ≤ 55 GeV. Also shown are those points that in addition satisfy the LUX limit.

Resonance region
In this subsection we focus on the h resonance region, 55 GeV < m S ≤ 70 GeV, which is defined such that the h is near the pole of SS annihilation, m S ≈ m h /2. In this region, the annihilation of SS into SM particles is mainly mediated through exchanging an s-channel h (unless the H is not much heavier than the h, m H ≈ m h ). For a given magnitude of λ h (and λ H when m H is close to m h ), the annihilation cross section is greatly enhanced in the resonance region, as seen in Fig 12 for Type I and figure 16 for Type II, respectively. In order to compensate for the resonance enhancement, λ h and/or λ H in the resonance region must be small in order to reproduce the observed DM abundance, as shown in figure 19. 8 In the upper plots of this figure, we have imposed the full set of preLUX constraints including BR(h → SS) ≤ 0.1. In the lower plots, we have required that the LUX limits also be obeyed. We observe that this latter requirement reduces further the magnitudes of λ h and λ H .
We also note that in this resonance region BR(H → SS) is typically large, between ∼ 0.1 and ∼ 0.9. If we were to repeat the plots of figure 18 for this case, we would find little change in the regions allowed, just an increase in point density. Indeed, very few of the starting 2HDM red points of figure 1 are eliminated by the preLUX constraints, implying that the regions shown are nearly identical to those for the original 2HDM points sampled. The reason for this is that once we are in the resonance region correct relic density can almost always be obtained by judiciously choosing λ h and λ H .

High mass region
The high mass region is defined as 70 ≤ m S ≤ 1000 GeV. In our study, the parameters κ 1 and κ 2 in the extra singlet sector are both scanned over in the range (10 −2 , 4π) with JHEP11(2014)105  figure 20. The lower plots show the points that also survive the LUX bound. There, one can observe that for high m S there is an ample parameter space surviving the preLUX constraints together with the LUX bound on the spin-independent cross section of DM direct detection.

Conclusions
We have analyzed the 2HDMS models obtained by extending the Type I and Type II two-Higgs-doublet models to include a scalar gauge-singlet dark matter candidate, denoted S with mass m S . We have discussed various theoretical and experimental constraints on the 2HDMS and how these constrain the additional (beyond the 2HDM) three parameters of JHEP11(2014)105 Figure 19. In the upper plots we show the couplings ouplings λ h , λ H allowed by the full set of preLUX constraints when m S is in the resonance mass region, 55 GeV < m S ≤ 70 GeV. In the lower plots, we show the points of the upper plots that are also consistent with the LUX limit on DM scattering. the 2HDMS, m S and the trilinear hSS and HSS couplings. We begin with the 2HDM fits of [32] for the case where it is the lighter h that is identified with the ∼ 125.5 GeV state, in particular employing the 2HDM parameter space points for which the combined LHC/Tevatron signal strengths are fit within the 68% C.L. We then study the constraints on the singlet parameter space based on cosmological data, most particularly the observed Ωh 2 and the LUX and SuperCDMS limits on DM-nucleon scattering. If m S > 55 GeV, 2HDMS parameter choices for which the 2HDMS is completely consistent with all the above data are plentiful in both the Type I and Type II models. For m S ≤ 55 GeV, requiring BR(h → SS) ≤ 0.1 in order to avoid destroying the fit of the h to the LHC data makes it impossible (almost impossible) in the Type I (Type II) model to find parameter points that give correct Ωh 2 and satisfy both the LUX and SuperCDMS limits. Nonetheless, it is interesting to note that if we do not impose the LUX and SuperCDMS limits, for both model types m S < 50 GeV-points with BR(h → SS) ≤ 0.1 and correct Ωh 2 fall within one or more of the CDMS II, CRESST-II or CoGeNT signal regions.
An important issue in the 2HDMS context is whether or not there is a possibility of isospin violation, f n /f p = 1. In the case of the 2HDMS Type I model, f n /f p 1 is inevitable. This, implies that despite the fact that all points with correct Ωh 2 and BR(h → JHEP11(2014)105 In the case of the 2HDMS Type II model, a significant isospin violation in DM-nucleon scattering is possible, even reaching the value of f n /f p ∼ −0.7 that would allow consistency of the LUX limit with the CDMS II signal region. However, at the low m S values corresponding to the signal region, we find that points with f n /f p ∼ −0.7 either have an hSS coupling that is too large for BR(h → SS) ≤ 0.1 or too small to give sufficient annihilation to achieve correct Ωh 2 . (At low m S , the H exchange contribution to SS annihilation is not sufficient, given upper bounds on the HSS coupling coming from perturbativity and unitarity.) Therefore, even though isospin violation might be present, the level of f n /f p ∼ −0.7 cannot be made consistent with all phenomenological requirements. The SuperCDMS limit further constrains the picture. For the f n /f p values predicted by the 2HDMS once correct Ωh 2 and BR(h → SS) ≤ 0.1 are imposed, the isospin violation is only a small effect in comparing the Germanium target SuperCDMS limit to the Silicon target CDMS II result. In the end, one does find a few m S ∼ 5.5 − 6.2 GeV-points that lie below both the SuperCDMS and LUX limits and, interestingly, also fall within (but are outside) the 99% C.L. (95% C.L.) CDMS II signal region. As typical for m S ≤ 50 GeV, these points are such that BR(H → SS) is large, implying that jet-and/or photon-tagging will be needed for H detection.

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This can be easily shown to be true using basic properties of a quadratic function. Lemma 1 for ξ = y leads to the following set of constraints: where A > 0 is the regular 2HDM constraint and B > 0 leads to λ S > 0. We can rewrite C > − √ 4A B in a way to use Lemma I again.