New LUX and PandaX-II Results Illuminating the Simplest Higgs-Portal Dark Matter Models

Direct searches for dark matter (DM) by the LUX and PandaX-II Collaborations employing xenon-based detectors have recently come up with the most stringent limits to date on the spin-independent elastic scattering of DM off nucleons. For Higgs-portal scalar DM models, the new results have precluded any possibility of accommodating low-mass DM as suggested by the DAMA and CDMS II Si experiments utilizing other target materials, even after invoking isospin-violating DM interactions with nucleons. In the simplest model, SM+D, which is the standard model plus a real singlet scalar named darkon acting as the DM candidate, the LUX and PandaX-II limits rule out DM masses roughly from 4 to 450 GeV, except a small range around the resonance point at half of the Higgs mass where the interaction cross-section is near the neutrino-background floor. In the THDM II+D, which is the type-II two-Higgs-doublet model combined with a darkon, the region excluded in the SM+D by the direct searches can be recovered due to suppression of the DM effective interactions with nucleons at some values of the ratios of Higgs couplings to the up and down quarks, making the interactions significantly isospin-violating. However, in either model, if the 125-GeV Higgs boson is the portal between the dark and SM sectors, DM masses less than 50 GeV or so are already ruled out by the LHC constraint on the Higgs invisible decay. In the THDM II+D, if the heavier $CP$-even Higgs boson is the portal, theoretical restrictions from perturbativity, vacuum stability, and unitarity requirements turn out to be important instead and exclude much of the region below 100 GeV. For larger DM masses, the THDM II+D has plentiful parameter space that corresponds to interaction cross-sections under the neutrino-background floor and therefore is likely to be beyond the reach of future direct searches without directional sensitivity.


I. INTRODUCTION
Cosmological studies have led to the inference that ordinary matter makes up only about 5% of the energy budget of the Universe, the rest being due to dark matter (26%) and dark energy (69%), the properties of which are largely still unknown [1]. Although the evidence for cosmic dark matter (DM) has been established for decades from numerous observations of its gravitational effects, the identity of its basic constituents has so far remained elusive. As the standard model (SM) of particle physics cannot account for the bulk of the DM, it is of great interest to explore various possible scenarios beyond the SM that can accommodate it. Amongst the multitudes of DM candidates that have been proposed in the literature, those classified as weakly interacting massive particles (WIMPs) are perhaps the leading favorites [1]. The detection of a WIMP is then essential not only for understanding the nature of the DM particle, but also for distinguishing models of new physics beyond the SM.
Many different underground experiments have been and are being performed to detect WIMPs directly by looking for the signatures of nuclear recoils caused by the collisions between the DM and nucleons. The majority of these searches have so far come up empty, leading only to upper bounds on the cross section σ N el of spin-independent elastic WIMP-nucleon scattering. Experiments utilizing xenon as the target material have turned out to supply the strictest bounds to date, especially the newest ones reported separately by the LUX and PandaX-II Collaborations [2,3], under the implicit assumption that the DM interactions with the proton and neutron respect isospin symmetry. These null results are in conflict with the tentative indications of WIMP signals observed earlier at relatively low masses in the DAMA [4] and CDMS II Si [5] measurements, which employed nonxenon target materials. 1 A graphical comparison between the new limits on σ N el from LUX and PandaX-II and the hypothetical signal regions suggested by DAMA and CDMS II Si is presented in Fig. 1(a). It also displays the limits from a few other direct searches [9][10][11], which were more sensitive to lighter WIMPs, as well as the expected reaches [12] of the upcoming XENON1T [13], DarkSide G2 [14], and LZ [15] experiments and an estimate of the WIMP discovery limit due to coherent neutrino scattering backgrounds [16].
Mechanisms that may reconcile the incompatible null and positive results of the WIMP DM direct searches have been suggested over the years. One of the most appealing proposals stems from the realization that the effective couplings f p and f n of the DM to the proton and neutron, respectively, may be very dissimilar [17,18]. If such a substantial violation of isospin symmetry occurs, the impact on the detection sensitivity to WIMP collisions can vary significantly, depending on the target material. In particular, during the collision process the DM may manifest a xenophobic behavior brought about by severe suppression of the collective coupling of the DM to xenon nuclei, but not necessarily to other nuclei [19]. This can explain why xenon-based detectors still have not discovered any DM, but DAMA and CDMS II Si perhaps did. Numerically, in the xenon case the suppression is the strongest if f n /f p ≃ −0.7 [18]. Assuming this ratio and applying it to the pertinent formulas provided in Ref. [18], one can translate the data in Fig. 1(a) into the corresponding numbers for the spin-independent elastic WIMP-proton cross-section, σ p el . confidence level (CL) versus WIMP mass from LUX [2], PandaX-II [3], CDMSlite [9], SuperCDMS [10], and CRESST [11] in the isospin-symmetric limit. Also shown are a gray patch compatible with the DAMA Na modulation signal at the 3σ level [20], a cyan area for the possible DM hint from CDMS II Si at 90% CL [5], the sensitivity projections [12] of XENON1T [13] (black dotted curve), DarkSide G2 [14] (maroon dash-dot-dotted curve), and LZ [15] (turquoise dash-dotted curve), and the WIMP discovery lower-limit due to coherent neutrino scattering backgrounds [16] (brown dashed curve). (b) The corresponding WIMP-proton cross-sections computed from (a) with isospin-violating effective WIMP couplings to the neutron and proton in the ratio f n /f p = −0.7.
The latter are plotted in Fig. 1(b), where the curve for DarkSide G2, which will employ an argon target, is scaled up differently from the curves for the xenon experiments including LZ. It is now evident that the conjectured signal regions of DAMA and CDMS II Si are no longer viable in light of the latest LUX and PandaX-II bounds. 2 Since these new results have reduced further the allowed WIMP parameter space, it is of interest to investigate what implications they may have for the simplest Higgs-portal WIMP DM models and how these scenarios may be probed more stringently in the future. For definiteness, in this paper we focus on the SM+D, which is the SM minimally expanded with the addition of a real singlet scalar serving as the DM and dubbed darkon, and on its two-Higgs-doublet extension of type II, which we call THDM II+D. 3 Specifically, we look at a number of constraints on these two models not only from the most recent DM direct searches, but also from LHC measurements on the gauge and Yukawa couplings of the 125-GeV Higgs boson and on its invisible decay mode, as well as from some theoretical requirements. We find that in the SM+D the darkon mass region up to ∼ 450 GeV is ruled out, except a small range near the resonant point at half of the Higgs mass where the DM-nucleon cross-section is close to the neutrino-background floor. On the other hand, in the THDM II+D the region excluded in the SM+D can be partially recovered because of suppression of the cross section that happens at some values of the product tan α tan β or cot α tan β, where α is the mixing angle of the CP -even Higgs bosons and tan β the ratio of vacuum expectation values (VEVs) of the Higgs doublets.
The structure of the rest of the paper is as follows. We treat the SM+D in Sec. II and the THDM II+D in Sec. III. We summarize our results and conclude in Sec. IV. A couple of appendices contain additional formulas and extra details.

II. CONSTRAINTS ON SM+D
The darkon, D, in the SM+D is a real scalar field and transforms as a singlet under the gauge group of the SM. Being the DM candidate, D is stable due to an exactly conserved discrete symmetry, Z 2 , under which D → −D, all the other fields being unaffected. The renormalizable darkon Lagrangian then has the form [23] where λ D , m 0 , and λ are free parameters and H is the Higgs doublet containing the physical Higgs field h. After electroweak symmetry breaking where the second term contains the darkon mass m D = m 2 0 +λv 2 1/2 , the last two terms play an important role in determining the DM relic density, and v ≃ 246 GeV is the vacuum expectation value (VEV) of H. Clearly, the darkon interactions depend on a small number of free parameters, the relevant ones here being the darkon-Higgs coupling λ, which pertains to the relic density, and the darkon mass m D .
In the SM+D, the relic density results from the annihilation of a darkon pair into SM particles which is induced mainly by the Higgs-exchange process DD → h * → X sm , where X sm includes all kinematically allowed final states at the darkon pair's center-of-mass (c.m.) energy, √ s. If the energy exceeds twice the Higgs mass, √ s > 2m h , the channel DD → hh also contributes, which arises from contact and (s, t, u)-channel diagrams. Thus, we can write the cross section σ ann of the darkon annihilation into SM particles as withh being a virtual Higgs having the same couplings as the physical h and an invariant mass equal to √ s, and the expression for σ(DD → hh) can be found in Appendix A, which also includes an outline of how λ is extracted from the observed abundance of DM. The resulting values of λ can then be tested with constraints from other experimental information.
In numerical work, we take m h = 125.1 GeV, based on the current data [32], and correspondingly the SM Higgs width Γ sm h = 4.08 MeV [33]. For m D < m h /2, the invisible decay channel h → DD is open and contributes to the Higgs' total width The Higgs measurements at the LHC provide information pertinent to this process. In the latest combined analysis on their Higgs data, the ATLAS and CMS Collaborations [34] have determined the branching fraction of h decay into channels beyond the SM to be B exp bsm = 0.00 +0. 16 , which can be interpreted as setting a cap on the Higgs invisible decay, B(h → invisible) exp < 0.16. Accordingly, we can impose which as we will see shortly leads to a major restriction on λ for m D < m h /2.
Direct searches for DM look for the nuclear recoil effects of DM scattering off a nucleon, N. In the SM+D, this is an elastic reaction, DN → DN, which is mediated by the Higgs in the t channel and has a cross section of for momentum transfers small relative to m h , where g N N h is the Higgs-nucleon effective coupling. Numerically, we adopt g N N h = 0.0011, which lies at the low end of our earlier estimates [28,35] and is comparable to other recent calculations [25,36]. The strictest limitations on σ N el to date are supplied by the newest null findings of LUX [2] and PandaX-II [3].
To show how these data confront the SM+D, we display in Fig. 2(a) the values of |λ| derived from the observed relic abundance (green solid curve) and compare them to the upper bounds on |λ| inferred from Eq. (5) based on the LHC information on the Higgs invisible decay [34] (black dotted curve) and from the new results of LUX [2] (red dashed curve) and PandaX-II [3] (orange dashed curve). The plot in Fig. 2(b) depicts the corresponding prediction for σ N el (green curve) in comparison to the same DM direct search data and future potential limits as in Fig. 1(a).
In the SM+D context, the graphs in Fig. 2 reveal that the existing data rule out darkon masses below about 450 GeV, except for the narrow dip area in the neighborhood of m D = m h /2, more precisely 52.1 GeV m D 62.6 GeV. At m D = m h /2, the threshold point for h → DD, the darkon annihilation into SM particles undergoes a resonant enhancement, and consequently a small size of λ can lead to the correct relic density and, at the same time, a low cross-section of darkon-nucleon collision. However, as Fig. 2 indicates, the bottom of the λ dip does not go to zero due to the Higgs' finite total width Γ h and the annihilation cross-section at the resonant point being proportional to 1/Γ 2 h . It is interesting to note that in Fig. 2(b) the bottom of the resonance region almost touches the expected limit of DM direct detection due to coherent neutrino scattering backgrounds. We also notice that the planned XENON1T, DarkSide G2, and LZ experiments [12] can probe the dip much further, but not all the way down. Thus, to exclude the dip completely a more sensitive machine will be needed. For darkon masses above 450 GeV, tests will be available from the ongoing PandaX-II as well as the forthcoming quests: particularly, XENON1T, DarkSide G2, and LZ can cover up to ∼ 3.5, 10, and a few tens TeV, respectively.

III. CONSTRAINTS ON THDM II+D
There are different types of the two-Higgs-doublet model (THDM), depending on how the two Higgs doublets, H 1 and H 2 , couple to SM fermions [37,38]. In the THDM I, only one of the doublets is responsible for endowing mass to all the fermions. In the THDM II, the up-type fermions get mass from only one of the Higgs doublets, say H 2 , and the down-type fermions from the other doublet. In the THDM III, both H 1 and H 2 give masses to all the fermions.
Since only one Higgs doublet generates all of the fermion masses in the THDM I, the couplings of each of the CP -even Higgs bosons to fermions are the same as in the SM, up to an overall scaling factor. Therefore, the couplings of the 125-GeV Higgs, h, in the THDM I slightly enlarged with the addition of a darkon are similar to those in the SM+D treated in the previous section, and consequently for m D < m h /2 the modifications cannot readily ease the restraints from the DM direct searches and LHC quest for the Higgs invisible decay. Combining a darkon with the THDM III instead could provide the desired ingredients to help overcome these obstacles [28], but the model possesses too many parameters to be predictable, some of which give rise to undesirable flavor-changing neutral-Higgs transitions at tree level. For these reasons, in the remainder of the section we concentrate on the THDM II plus the darkon (THDM II+D).
In the THDM II+D, the fermion sector is no different from that in the THDM II, with the Yukawa interactions being described by [37,38] where summation over j, l = 1, 2, 3 is implicit, Q j,L (L j,L ) represents left-handed quark (lepton) doublets, U l,R and D l,R (E l,R ) denote right-handed quark (charged lepton) fields, with τ 2 being the second Pauli matrix, and λ u,d,ℓ are 3×3 matrices for the Yukawa couplings. This Lagrangian respects the discrete symmetry, Z 2 , under which H 2 → −H 2 and U R → −U R , while all the other fields are not affected. Thus, and their Hermitian conjugates from occurring in L Y .
The longevity of the darkon as the DM in the THDM II+D is maintained by another discrete symmetry, Z ′ 2 , under which D → −D, whereas all the other fields are Z ′ 2 even. Consequently, being a real field and transforming as a singlet under the SM gauge group, D has no renormalizable interactions with SM fermions or gauge bosons, like in the SM+D.
The renormalizable Lagrangian of the model, L ⊃ −V D − V H , contains the scalar potential terms [27] where V H is the usual THDM II potential [37,38]. Because of Z 2 , the combinations and their Hermitian conjugates are forbidden from appearing in Eq. (8). However, in V H we have included the m 2 12 terms which softly break Z 2 and are important in relaxing the upper bounds on the Higgs masses [38]. In contrast, Z ′ 2 , which guarantees the darkon stability, is exactly conserved. The Hermiticity of V D,H implies that the parameters m 2 0,11,22 and λ D,1D,2D,1,2,3,4 are real. We assume V D,H to be CP invariant, and so m 2 12 and λ 5 are also real parameters.
The λ 1D,2D terms in Eq. (8) play a crucial role in the determination of the relic density, which follows from darkon annihilation into the other particles via interactions with the Higgs bosons. To address this in more detail, we first decompose the Higgs doublets as where v 1,2 are the VEVs of H 1,2 , respectively, and connected to the electroweak scale v ≃ 246 GeV by v 1 = v cos β and v 2 = v sin β. The H r components h + r , h 0 r , and I 0 r are related to the physical Higgs bosons h, H, A, and H + and the would-be Goldstone bosons w + and z by where X is any angle or combination of angles. The w ± and z will be eaten by the W ± and Z bosons, respectively.
After electroweak symmetry breaking, we can then express the relevant terms in V = V D + V H involving the physical bosons as where and the cubic couplings λ XYZ are listed in Appendix A. There is no AD 2 term under the assumed CP invariance. Since m 0 and λ 1,2 are free parameters, so are m D and λ h,H . The quartic couplings of the darkon to the Higgs bosons can then be related to λ h,H by Since h and H couple directly to the weak bosons, we need to include the annihilation channels DD → W + W − , ZZ if kinematically permitted. The pertinent interactions are given by The scattering of the darkon off a nucleon N = p or n is generally mediated at the quark level by h and H and hence depends not only on the darkon-Higgs couplings λ h,H , but also on the effective Higgs-nucleon coupling g N N H defined by This originates from the quark-Higgs terms in Eq. (7) given by where the sum is over all quarks, q = u, d, s, c, b, t, and It follows that [39] where f N q is defined by the matrix element N |m q qq|N = f N q m N u N u N with u N being the Dirac spinor for N and m N its mass. Employing the values f N q for the different quarks listed in Appendix A, we find Setting k h u,d = 1 in these formulas, we reproduce the SM values g sm pph,nnh ≃ 0.0011 quoted in the last section. However, if k H u,d are not close to unity, g ppH and g nnH can be very dissimilar, breaking isospin symmetry substantially. Particularly, they have different zeros, respectively. This suggests that to evaluate DM collisions with nucleons in the THDM II+D it is more appropriate to work with either the darkon-proton or darkon-neutron cross-section (σ p el or σ n el , respectively) rather than the darkon-nucleon one under the assumption of isospin conservation. The calculated σ p,n el can then be compared to their empirical counterparts which are converted from the measured σ N el using the relations [18,19] where the sums are over the isotopes of the element in the target material with which the DM interacts dominantly, η i (A i ) represent the fractional abundances (the nucleon numbers) of the isotopes, with m A i being the ith isotope's mass, Z denotes the proton number of the element, and f n /f p is fixed under certain assumptions. For illustration, from Eq. (20) we graph σ N el /σ p el as a function f n /f p for a few target materials (silicon, argon, and xenon) in Fig. 3, where the curves are not sensitive to the darkon masses in our range of interest. Thus, if there is no isospin violation, f n = f p leading to σ p el = σ N el . On the other hand, for DM with maximal xenophobia, f n /f p = −0.70, and with this number we arrived at Fig. 1(b) from Fig. 1(a). More generally, σ p el can be bigger or smaller than σ N el if f n = f p , but completely destructive interference on the right-hand side of the first relation in Eq. (20) yielding σ N el /σ p el = 0 is not achievable if the element has more than one naturally abundant isotope. If both the h and H couplings to the darkon are nonzero, the cross section of the darkon-N scattering DN → DN is for momentum transfers small relative to m h,H and N = p or n. Given that g N N H depends on k H u,d according to Eq. (18), it may be possible to make g N N H sufficiently small with a suitable choice of k H d /k H u to allow σ N el to avoid its experimental limit [29], at least for some of the m D values. Moreover, the λ h,H terms in Eq. (21) may (partially) cancel each other to reduce σ N el as well. These are attractive features of the THDM II+D that the SM+D does not possess.
Since there are numerous different possibilities in which h and H may contribute to darkon interactions with SM particles in the THDM+D, hereafter for definiteness and simplicity we focus on a couple of scenarios in which h is the 125-GeV Higgs boson and the other Higgs bosons are heavier, m h < m H,A,H ± . In addition, we assume specifically that either H or h has a vanishing coupling to the darkon, λ H = 0 or λ h = 0, respectively. As a consequence, either h or H alone serves as the portal between the DM and SM particles, and so we now have f n /f p = g nnH /g ppH , upon neglecting the n-p mass difference.
If we take g nnH /g ppH = −0.70, which corresponds to the xenophobic limit, using Eq. (19) we get r H k ≡ k H d /k H u = −0.96, where r h k = − tan α tan β and r H k = cot α tan β from Eq. (17). Nevertheless, as we see later on, despite the strongest constraints to date from xenon-based detectors, higher r H k values are still compatible with the data and hence the darkon can still avoid extreme xenophobia. The choices for α and β, however, need to comply with further restraints on k h d,u,V , as specified below. Given that LHC measurements have been probing the Higgs couplings to SM fermions and electroweak bosons, we need to take into account the resulting restrictions on potential new physics in the couplings. A modification to the h → XX interaction with respect to its SM expectation can be parameterized by κ X defined by κ 2 X = Γ h→XX /Γ sm h→XX . Assuming that |κ W,Z | ≤ 1 and the Higgs total width can get contributions from decay modes beyond the SM, the ATLAS and CMS Collaborations have performed simultaneous fits to their Higgs data to extract [34] κ W = 0.90 ± 0.09 , κ t = 1.43 +0.23 −0.22 , |κ b | = 0.57 ± 0.16 , |κ γ | = 0.90 +0.10 −0.09 , κ Z = 1.00 −0.08 , |κ g | = 0.81 +0.13 −0.10 , |κ τ | = 0.87 +0.12 −0.11 , where [34] κ 2 γ = 0.07 κ 2 t +1.59 κ 2 W −0.66 κ t κ W . In the THDM II context, we expect these numbers to respect within one sigma the relations k h V = κ W = κ Z , k h u = κ t ≃ κ g , and k h d = κ b = κ τ , although the κ t,g (κ b,τ ) numbers above overlap only at the two-sigma level. Accordingly, pending improvement in the precision of these parameters from future data, based on Eq. (22) we may impose where k h γ incorporates the loop contribution of H ± to h → γγ, and so k h γ → κ γ if the impact of H ± is vanishing. Explicitly where A γγ 0 is a loop function whose expression can be found in the literature (e.g., [41]). The effect of the λ hH + H − term in k h γ turns out to be somewhat minor in our examples. To visualize the impact of the limitations in Eq. (23), we plot in Fig. 4 the (red) regions representing the α and β parameter space satisfying them.
Before proceeding to our specific scenarios of choice, we remark that in the alignment limit, β = α + π/2, we recover the SM+D darkon parameters, with λ h = λ. Furthermore, in this limit the h couplings become SM-like,

A. λ H = 0
In this case, the cross section of the darkon annihilation into THDM particles is where the first term on the right-hand side is equal to its SM+D counterpart in Eq. (3), except λ is replaced by λ h and the h couplings to fermions and gauge bosons are multiplied by the relevant k h u,d,V factors mentioned earlier, and the sum is over s 1 s 2 = hh, hH, HH, AA, H + H − with only kinematically allowed channels contributing. The formulas for σ(DD → s 1 s 2 ) have been relegated to Appendix A. Hence, though not the portal between the DM and SM particles in this scenario, H can still contribute to the darkon relic abundance via DD → s 1 s 2 , along with A and H ± .
Once λ h has been extracted from the relic density data and g N N h calculated with the α and β choices consistent with Eq. (17), we can predict the darkon-N cross-section. From now on, we work exclusively with the darkon-proton one, This is to be compared to its empirical counterparts derived from the σ N el data using Eq. (20) with f n /f p = g nnh /g pph . There are other restrictions that we need to take into account.
As in the SM+D, for m D < m h /2 the invisible channel h → DD is open and has a rate given by Eq. (4), with λ being replaced by λ h . The branching fraction of h → DD must then be consistent with the LHC measurement on the Higgs invisible decay, and so for this darkon mass range we again impose the bound in Eq. (5).
Since the extra Higgs particles in the THDM exist due to the second doublet being present, they generally affect the so-called oblique electroweak parameters S and T which encode the impact of new physics coupled the standard SU(2) L gauge boson [42]. Thus the new scalars must also comply with the experimental constraints on these quantities. To ensure this, we employ the pertinent formulas from Ref. [43] and the S and T data from Ref. [1].
Lastly, the parameters of the scalar potential V = V D + V H in Eq. (8) need to fulfill a number of theoretical conditions. The quartic couplings in V cannot be too big individually, for otherwise the theory will no longer be perturbative. Another requirement is that V must be stable, implying that it has to be bounded from below to prevent it from becoming infinitely negative for arbitrarily large fields. It is also essential to ensure that the (tree level) amplitudes for scalar-scalar scattering at high energies do not violate unitarity constraints. We address these conditions in more detail in Appendix B. They can be consequential in restraining parts of the model parameter space, especially for m D less than O(100 GeV), as some of our examples will later demonstrate.
To illustrate the viable parameter space in this scenario, in the second to seventh columns of Table I we put together a few sample sets of input parameters which are consistent with Eq. (23) and the requirements described in the last two paragraphs. The eighth to twelfth columns contain the resulting values of several quantities. With the input numbers from Set 1 in the table, we show in Fig. 5(a) the λ h region evaluated from the observed relic density. We also display the upper limits on λ h inferred from Eq. (5) for the h → DD limit (black dotted curve), from the latest LUX [2] and PandaX-II [3] searches, and from the aforementioned theoretical demands for perturbativity, potential stability, and unitarity.
The plot in Fig. 5(b) exhibits the corresponding prediction for σ p el (green curve) compared to its empirical counterparts obtained from the data depicted in Fig. 1(a) by employing Eq. (20) with f n /f p = 0.658 from Set 1 in Table I. One observes that the m D < 50 GeV region, represented by the dotted section of the green curve, is incompatible with the LHC constraint on h → DD and a portion of it is also excluded by LUX and PandaX-II. The green solid curve is below all of the existing limits from direct searches and for a narrow range of m D lies not far under the LUX line. Upcoming quests with XENON1T as well as DarkSide G2 will apparently be sensitive to only a small section of the green solid curve, below 100 GeV, whereas LZ can expectedly reach more of it, from about 63 to 170 GeV.  Table I. Also plotted are upper limits from the theoretical conditions mentioned in the text (horizontal purple dotted-line), the LHC Higgs invisible decay data (black dotted-curve), and the latest LUX (red dashed-curve) and PandaX-II (orange dashed-curve) results. (b) The corresponding darkon-proton cross-section σ p el (green curve), compared to its counterparts translated from the σ N el data and projections in Fig. 1(a) using Eq. (20) with f n /f p from Set 1 in Table I. The dotted portion of the green curve is excluded by the LHC bound in (a). Fig. 6 we graph analogous results with the input numbers from Sets 2 and 3 in Table I. Their f n /f p values are lower than that in Set 1, making the darkon more xenophobic and therefore harder to discover with xenon-based detectors, as can also be deduced from Fig. 3. Especially, in these instances the predictions for σ p el (green solid curves) are far less than the available experimental bounds and may be out of reach for direct searches in the not-too-distant future.

For further illustrations, in
For a more straightforward comparison between the model predictions and direct search results, which are typically reported in terms of the DM-nucleon cross-section σ N el , we have converted the calculated σ p el in Figs. 5 and 6 to the three (green) σ N el curves in Fig. 7(a) using Eq. (20) with the f n /f p values from Table I and assuming that the target material in the detector is xenon. Recalling that the DarkSide G2 experiment will employ an argon target [14], we plot the corresponding predictions for σ N el assuming an argon target instead in Fig. 7(b), which reveals some visible differences from Fig. 7(a) in the predictions with f n /f p < 0, as Fig. 3 would imply as well. Also shown are the same data and projections as in Fig. 1(a). From Fig. 7, we can conclude that near-future direct detection experiments will be sensitive to only a rather limited part of the h-portal THDM II+D parameter space. We notice specifically that the predicted σ N el in much of the m D > 100 GeV region is under the neutrino-background floor.  Table I.  Table I for (a) xenon and (b) argon targets as described in the text, compared to the same data and projections as in Fig. 1(a). The dotted portions of the green curves are excluded as in Figs. 5 and 6.

B. λ h = 0
In this scenario, the cross section of the darkon annihilation into THDM particles is where withH being a virtual H having the same couplings as the physical H and an invariant mass equal to √ s, and the sum in σ ann is again over s 1 s 2 = hh, hH, HH, AA, H + H − . For the H-mediated darkon-proton scattering, Dp → Dp, the cross section is In applying Eq. (20), we set f n /f p = g nnH /g ppH .
Similarly to the λ H = 0 case, here we present three examples, with their respective sets of input numbers being collected in Table II. We also impose the requirements described earlier in this section, except that the LHC information on the decay mode h → invisible is not useful for bounding λ H . Nevertheless, the theoretical conditions for perturbativity, stability of the potential, and unitarity of high-energy scalar scattering amplitudes turn out to be consequential in disallowing darkon masses less than 100 GeV. Table II lead to the graphs in Fig. 8 (9). In these figures, we see that the |λ H | values extracted from the relic density data tend to be bigger than their λ h counterparts in the λ H = 0 instances. This is because the H-mediated   in the λ h = 0 scenario and the resulting values of several quantities, including f n /f p = g nnH /g ppH .  Table II. (b) The corresponding darkon-proton cross-section σ p el (green curve), compared to its counterparts translated from Fig. 1(a) using Eq. (20) with f n /f p from Set 1 in Table II. The dotted portion of the green curve is excluded by the theoretical bound in (a).

The input numbers from Set 1 (Sets 2 and 3) in
annihilation rate is relatively more suppressed due to m H > m h . As a consequence, more of the low-m D regions are in conflict with the restrictions from the aforementioned theoretical requirements. Furthermore, in Fig. 8(b), like in Fig. 5(b), there is a small range of the solid green curve, around its leftmost end, that is close to the LUX and PandaX-II limits.
As in the previous subsection, assuming xenon to be the target material, we have translated the predicted σ p el in Figs. 8 and 9 into the three (green) σ N el curves in Fig. 10(a) in order to provide a more direct comparison with experimental results. If the target is argon instead and f n /f p < 0, the σ N el predictions can be visibly greater, as depicted in Fig. 10(b). For darkon masses above 100 GeV, the majority of the σ N el predictions in Fig. 10 appear to lie under the neutrino-background floor, more in these instances than those in the λ H = 0 case. Thus, the λ h = 0 scenario is likely to be comparatively more challenging to probe with direct searches.  Table II.  Table II for (a) xenon and (b) argon targets as described in the text, compared to the same data and projections as in Fig. 1(a). The dotted portions of the green curves are excluded as in Figs. 8 and 9.

IV. CONCLUSIONS
We have explored some of the implications of the most recent null results of WIMP DM direct searches by LUX and PandaX-II. For Higgs-portal scalar WIMP DM models, the new limits have eliminated any possibility to accommodate low-mass DM undergoing spin-independent elastic scattering off nucleons that was suggested by the potentially positive results of the DAMA and CDMS II Si experiments, even after invoking the mechanism of isospin violation in DM-nucleon interactions. We have studied particularly how the LUX and PandaX-II results probe the parameter space of the simplest Higgs-portal scalar DM models, namely the SM+D, which is the SM plus a real scalar singlet called darkon, and the THDM II+D, which is the two-Higgs-doublet model of type II combined with a darkon. In the THDM II+D we entertain the possibility that the 125-GeV Higgs boson, h, is the lightest one of the physical members of the scalar doublets. Our analysis takes into account various constraints from LHC data on the Yukawa couplings of h, its couplings to gauge bosons, and its invisible decay mode. Also pertinent are restrictions from oblique electroweak precision measurements and from theoretical considerations regarding perturbativity, vacuum stability, and unitarity. In the SM+D case, h is the only portal between the DM and SM sectors, while in the THDM II+D one or both of the CP -even Higgs bosons, h and the heavier H, can be the portals.
We find that in scenarios with h being the only portal the LHC information on h → invisible places a significant restraint on the darkon-h coupling and rules out the m D ≤ m h /2 region, except a small range near the resonance point m D = m h /2. We also find that for m D > m h /2 in the SM+D the new LUX and PandaX-II limits exclude masses up to 450 GeV or so, but in the h-portal THDM II+D they can be recovered due to suppression of the Higgs-nucleon coupling, g N N h , at some values of the product tan α tan β. In contrast, in the THDM II+D scenario with H being the sole portal, the h → invisible bound does not apply to the much heavier H, and the LUX and PandaX-II limits can be evaded due to suppression of g N N H at some values of cot α tan β. However, in this case our examples demonstrate that the foregoing theoretical requirements are consequential and disallow most of the m D < 100 GeV region. Thus, darkon masses below m D ≃ 50 GeV are ruled out in the SM+D by LHC data and very likely so in the THDM II+D by the LHC and theoretical restrictions. For higher masses, lower parts of the dip around m D = m h /2 in the h-portal cases will remain viable for the foreseeable future, and beyond the h-resonance area the region up to roughly 3.5, 10, and 20 TeV in the SM+D will be testable by XENON1T, DarkSide G2, and LZ, respectively. For m D > 100 GeV in the THDM II+D there is generally ample parameter space that yields a darkon-nucleon cross-section below the neutrino-background floor and is therefore likely to elude direct detection experiments in the future which lack directional sensitivity. Finally, we point out that the considerable suppression of g N N H is accompanied by g ppH and g nnH manifesting sizable isospin breaking, as illustrated in our examples.