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

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

Xiao-Gang He    Jusak Tandean INPAC, Department of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dongchuan Rd., Minhang, Shanghai 200240, China
Department of Physics and Center for Theoretical Sciences, National Taiwan University,
No.  1, Sec.  4, Roosevelt Rd., Taipei 106, Taiwan
Physics Division, National Center for Theoretical Sciences, No.  101, Sec.  2, Kuang Fu Rd., Hsinchu 300, Taiwan
###### Abstract

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 -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 pdg (). 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 pdg (). 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 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 lux (); pandax (), 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 dama () and CDMS II Si cdmssi () measurements, which employed nonxenon target materials.111The excess events previously observed in the CoGeNT cogent () and CRESST-II cresst () experiments have recently been demonstrated to be entirely attributable to underestimated backgrounds instead of DM recoils nosignal (). A  graphical comparison between the new limits on 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 Agnese:2014aze (); Agnese:2015nto (); Angloher:2015ewa (), which were more sensitive to lighter WIMPs, as well as the expected reaches Cushman:2013zza () of the upcoming XENON1T xenon1t (), DarkSide G2 dsg2 (), and LZ lz () experiments and an estimate of the WIMP discovery limit due to coherent neutrino scattering backgrounds nubg ().

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 and of the DM to the proton and neutron, respectively, may be very dissimilar Kurylov:2003ra (); Feng:2011vu (). 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 Feng:2013fyw (). 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    Feng:2011vu (). Assuming this ratio and applying it to the pertinent formulas provided in Ref. Feng:2011vu (), one can translate the data in Fig.  1(a) into the corresponding numbers for the spin-independent elastic WIMP-proton cross-section, . 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.222If the DM-nucleon scattering is both isospin violating and inelastic, which can happen if a spin-1 particle, such as a boson, is the portal between the DM and SM particles, it may still be possible to accommodate the potential hint of low-mass DM from CDMS II Si and evade the limits from xenon detectors at the same time inelastic (). The inelastic-DM approach has also been proposed to explain the DAMA anomaly Scopel:2015eoh ().

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.333There are earlier studies in the literature on various aspects of the SM plus singlet scalar DM, or a greater scenario containing the model, in which the scalar was real Silveira:1985rk (); sm+reald (); Cline:2013gha () or complex sm+complexd (). Two-Higgs-doublet extensions of the SM+D have also been explored previously He:2008qm (); He:2011gc (); Bird:2006jd (); 2hdm+d (); Drozd:2014yla (). 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    or  ,  where is the mixing angle of the -even Higgs bosons and 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, , 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, is stable due to an exactly conserved discrete symmetry, , under which  ,  all the other fields being unaffected. The renormalizable darkon Lagrangian then has the form Silveira:1985rk ()

 LD=12∂μD∂μD−14λDD4−12m20D2−λD2H†H, (1)

where , , and are free parameters and is the Higgs doublet containing the physical Higgs field . After electroweak symmetry breaking

 (2)

where the second term contains the darkon mass  ,  the last two terms play an important role in determining the DM relic density, and   GeV  is the vacuum expectation value (VEV) of . Clearly, the darkon interactions depend on a small number of free parameters, the relevant ones here being the darkon-Higgs coupling , which pertains to the relic density, and the darkon mass .

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  ,  where includes all kinematically allowed final states at the darkon pair’s center-of-mass (c.m.) energy, . If the energy exceeds twice the Higgs mass,  ,  the channel    also contributes, which arises from contact and -channel diagrams. Thus, we can write the cross section of the darkon annihilation into SM particles as

 σann = σ(DD→h∗→X\textscsm)+σ(DD→hh), σ(DD→h∗→X\textscsm) = 4λ2v2(m2h−s)\raisebox1.0pt$2$+Γ2hm2h ∑iΓ(~h→Xi,\textscsm)√s−4m2D,       X\textscsm≠hh, (3)

with being a virtual Higgs having the same couplings as the physical and an invariant mass equal to , and the expression for can be found in Appendix  A, which also includes an outline of how is extracted from the observed abundance of DM. The resulting values of can then be tested with constraints from other experimental information.

In numerical work, we take   GeV,  based on the current data lhc:mh (), and correspondingly the SM Higgs width   MeV  lhctwiki (). For  ,  the invisible decay channel    is open and contributes to the Higgs’ total width    in Eq. (3), where

 Γ(h→DD)=λ2v28πmh ⎷1−4m2Dm2h. (4)

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 atlas+cms () have determined the branching fraction of decay into channels beyond the SM to be  ,  which can be interpreted as setting a cap on the Higgs invisible decay,  .  Accordingly, we can impose

 B(h→DD)=Γ(h→DD)Γh<0.16, (5)

which as we will see shortly leads to a major restriction on for  .

Direct searches for DM look for the nuclear recoil effects of DM scattering off a nucleon, . In the SM+D, this is an elastic reaction,  ,  which is mediated by the Higgs in the channel and has a cross section of

 σNel=λ2g2NNhm2Nv2π(mD+mN)2m4h (6)

for momentum transfers small relative to , where is the Higgs-nucleon effective coupling. Numerically, we adopt  ,  which lies at the low end of our earlier estimates He:2010nt (); He:2011gc () and is comparable to other recent calculations Cline:2013gha (); Cheng:2012qr (). The strictest limitations on to date are supplied by the newest null findings of LUX lux () and PandaX-II pandax ().

To show how these data confront the SM+D, we display in Fig. 2(a) the values of derived from the observed relic abundance (green solid curve) and compare them to the upper bounds on inferred from Eq. (5) based on the LHC information on the Higgs invisible decay atlas+cms () (black dotted curve) and from the new results of LUX lux () (red dashed curve) and PandaX-II pandax () (orange dashed curve). The plot in Fig. 2(b) depicts the corresponding prediction for (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  ,  more precisely  52.1  GeV 62.6  GeV.  At  ,  the threshold point for  ,  the darkon annihilation into SM particles undergoes a resonant enhancement, and consequently a  small size of 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 dip does not go to zero due to the Higgs’ finite total width and the annihilation cross-section at the resonant point being proportional to . 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 Cushman:2013zza () 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, and , couple to SM fermions thdm (); Branco:2011iw (). 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 , and the down-type fermions from the other doublet. In the THDM  III, both and 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 -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, , 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    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 He:2011gc (), 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 thdm (); Branco:2011iw ()

 LY=−¯¯¯¯Qj,L(λu2)jl~H2Ul,R−¯¯¯¯Qj,L(λd1)jlH1Dl,R−¯¯¯¯Lj,L(λℓ1)jlH1El,R+H.c., (7)

where summation over    is implicit, represents left-handed quark (lepton) doublets,   and denote right-handed quark (charged lepton) fields,    with being the second Pauli matrix, and are 33 matrices for the Yukawa couplings. This Lagrangian respects the discrete symmetry, , under which    and  ,  while all the other fields are not affected. Thus, prohibits the combinations  , , , and their Hermitian conjugates from occurring in .

The longevity of the darkon as the DM in the THDM II+D is maintained by another discrete symmetry, , under which  ,  whereas all the other fields are even. Consequently, being a real field and transforming as a singlet under the SM gauge group, has no renormalizable interactions with SM fermions or gauge bosons, like in the SM+D.

The renormalizable Lagrangian of the model,  ,  contains the scalar potential terms Bird:2006jd ()

 VD = VH = m211H†1H1+m222H†2H2−(m212H†1H2+H.c.)+λ12(H†1H1)\raisebox1.0pt$2$+λ22(H†2H2)\raisebox1.0pt$2$ (8) +λ3H†1H1H†2H2+λ4H†1H2H†2H1+λ52[(H†1H2)\raisebox1.0pt$2$+H.c.],

where is the usual THDM  II potential thdm (); Branco:2011iw (). Because of , the combinations , , , and their Hermitian conjugates are forbidden from appearing in Eq. (8). However, in we have included the terms which softly break and are important in relaxing the upper bounds on the Higgs masses Branco:2011iw (). In contrast, , which guarantees the darkon stability, is exactly conserved. The Hermiticity of implies that the parameters and are real. We assume to be invariant, and so and are also real parameters.

The 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

 Hr=1√2(√2h+rvr+h0r+iI0r),     r=1,2, (9)

where are the VEVs of , respectively, and connected to the electroweak scale   GeV  by    and  .  The components , , and are related to the physical Higgs bosons , , , and and the would-be Goldstone bosons and by

 (h+1h+2) = (cβ−sβsβcβ)(w+H+),       (I01I02)=(cβ−sβsβcβ)(zA), (h01h02) = (cα−sαsαcα)(Hh),cX=cosX,     sX=sinX, (10)

where is any angle or combination of angles. The and will be eaten by the and bosons, respectively.

After electroweak symmetry breaking, we can then express the relevant terms in    involving the physical bosons as

 V ⊃ 12m2DD2+(λhh+λHH)D2v (11) +12(λhhh2+2λhHhH+λHHH2+λAAA2+2λH+H−H+H−)D2 +(16λhhhh2+12λhhHhH+12λhHHH2+12λhAAA2+λhH+H−H+H−)hv +(16λHHHH2+12λHAAA2+λHH+H−H+H−)Hv,

where  ,

 λh = λ2Dcαsβ−λ1Dsαcβ,λH=λ1Dcαcβ+λ2Dsαsβ, λhh = λ1Ds2α+λ2Dc2α,λHH=λ1Dc2α+λ2Ds2α, λhH = (λ2D−λ1D)cαsα,λAA=λH+H−=λ1Ds2β+λ2Dc2β, (12)

and the cubic couplings are listed in Appendix  A. There is no term under the assumed invariance. Since and are free parameters, so are and . The quartic couplings of the darkon to the Higgs bosons can then be related to by

 λhh = (c3αsβ−s3αcβ)λh+s2αcβ−αs2βλH,λHH=(c3αcβ+s3αsβ)λH−s2αsβ−αs2βλh, λhH = s2αs2β(λhcβ−α−λHsβ−α),λAA=λH+H−=cαc3β−sαs3βcβsβλh+cαs3β+sαc3βcβsβλH. (13)

Since and couple directly to the weak bosons, we need to include the annihilation channels    if kinematically permitted. The pertinent interactions are given by

 L⊃(2m2WW+μW−μ+m2ZZμZμ)(khVhv+kHVHv),       khV=sβ−α,     kHV=cβ−α. (14)

The scattering of the darkon off a nucleon   or   is generally mediated at the quark level by and and hence depends not only on the darkon-Higgs couplings , but also on the effective Higgs-nucleon coupling   defined by

 LNNH=−gNNH¯¯¯¯¯NNH,       H=h,H. (15)

This originates from the quark-Higgs terms in Eq. (7) given by

 LY⊃−\raisebox−7.0pt$\lx@stackrel∑\scriptsizeq$kHqmq¯¯¯qqHv,       kHc,t=kHu,     kHs,b=kHd, (16)

where the sum is over all quarks,  ,  and

 khu=cαsβ,       khd=−sαcβ,       kHu=sαsβ,       kHd=cαcβ. (17)

It follows that Shifman:1978zn ()

 gNNH=mNv[(fNu+fNc+fNt)kHu+(fNd+fNs+fNb)kHd], (18)

where is defined by the matrix element    with being the Dirac spinor for and its mass. Employing the values for the different quarks listed in Appendix  A, we find

 gppH=(0.5631kHu+0.5599kHd)×10−3,     gnnH=(0.5481kHu+0.5857kHd)×10−3. (19)

Setting    in these formulas, we reproduce the SM values    quoted in the last section. However, if are not close to unity, and can be very dissimilar, breaking isospin symmetry substantially. Particularly, they have different zeros,    and  ,  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 ( or , respectively) rather than the darkon-nucleon one under the assumption of isospin conservation. The calculated can then be compared to their empirical counterparts which are converted from the measured using the relations Feng:2011vu (); Feng:2013fyw ()

 (20)

where the sums are over the isotopes of the element in the target material with which the DM interacts dominantly, represent the fractional abundances (the nucleon numbers) of the isotopes,444A recent list of isotopic abundances can be found in isotopes ().,  with being the th isotope’s mass, denotes the proton number of the element, and is fixed under certain assumptions. For illustration, from Eq. (20) we graph as a function 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,    leading to  .  On the other hand, for DM with maximal xenophobia,  ,  and with this number we arrived at Fig. 1(b) from Fig. 1(a). More generally, can be bigger or smaller than   if  ,  but completely destructive interference on the right-hand side of the first relation in Eq. (20) yielding    is not achievable if the element has more than one naturally abundant isotope.

If both the and couplings to the darkon are nonzero, the cross section of the darkon- scattering    is

 σNel=m2Nv2π(mD+mN)2(λhgNNhm2h+λHgNNHm2H)2 (21)

for momentum transfers small relative to and   or .  Given that depends on according to Eq. (18), it may be possible to make sufficiently small with a suitable choice of to allow to avoid its experimental limit He:2008qm (), at least for some of the values. Moreover, the terms in Eq. (21) may (partially) cancel each other to reduce 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 and 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 is the 125-GeV Higgs boson and the other Higgs bosons are heavier,  .  In addition, we assume specifically that either or has a  vanishing coupling to the darkon,    or  ,  respectively. As a consequence, either or alone serves as the portal between the DM and SM particles, and so we now have  ,  upon neglecting the - mass difference.

If we take  ,  which corresponds to the xenophobic limit, using Eq. (19) we get  ,  where    and    from Eq. (17). Nevertheless, as we see later on, despite the strongest constraints to date from xenon-based detectors, higher values are still compatible with the data and hence the darkon can still avoid extreme xenophobia. The choices for and , however, need to comply with further restraints on , 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    interaction with respect to its SM expectation can be parameterized by defined by  .  Assuming that    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 atlas+cms ()

 κW = 0.90±0.09,       κt=1.43+0.23−0.22,       |κb|=0.57±0.16,       |κγ|=0.90+0.10−0.09, κZ = 1.00−0.08,        |κg|=0.81+0.13−0.10,      |κτ|=0.87+0.12−0.11, (22)

where atlas+cms ().  In the THDM  II context, we expect these numbers to respect within one sigma the relations  ,  ,  and  ,  although the numbers above overlap only at the two-sigma level. Accordingly, pending improvement in the precision of these parameters from future data, based on Eq. (III) we may impose

 0.81≤khV≤1,     0.71≤khu≤1.66,     0.41≤∣∣khd∣∣≤0.99,     0.81≤∣∣khγ∣∣≤1, (23)

where incorporates the loop contribution of to  ,  and so    if the impact of is vanishing. Explicitly

 khγ=0.264khu−1.259khV+0.151λhH+H−v22m2H±Aγγ0(4m2H±/m2h), (24)

where is a loop function whose expression can be found in the literature (e.g., Chen:2013vi ()). The effect of the term in 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 and parameter space satisfying them.

Before proceeding to our specific scenarios of choice, we remark that in the alignment limit,  ,  we recover the SM+D darkon parameters,

 m2D=m20+λhv2,       λhh=λh (25)

with  .  Furthermore, in this limit the couplings become SM-like,

 λhhh=3m2hv2,       khV=1,       khq=1. (26)

### iii.1 λH=0

In this case, the cross section of the darkon annihilation into THDM particles is

 σann=σ(DD→h∗→X\textscsm)+\raisebox−7.0pt$\lx@stackrel∑\scriptsizes1s2$σ(DD→s1s2), (27)

where the first term on the right-hand side is equal to its SM+D counterpart in Eq. (3), except is replaced by and the couplings to fermions and gauge bosons are multiplied by the relevant factors mentioned earlier, and the sum is over    with only kinematically allowed channels contributing. The formulas for have been relegated to Appendix  A. Hence, though not the portal between the DM and SM particles in this scenario, can still contribute to the darkon relic abundance via  ,  along with and .

Once has been extracted from the relic density data and calculated with the and choices consistent with Eq. (17), we can predict the darkon- cross-section. From now on, we work exclusively with the darkon-proton one,

 σpel=λ2hg2pphm2pv2π(mD+mp)\raisebox1.0pt$2$m4h. (28)

This is to be compared to its empirical counterparts derived from the data using Eq. (20) with  . There are other restrictions that we need to take into account.

As in the SM+D, for    the invisible channel    is open and has a rate given by Eq. (4), with being replaced by . The branching fraction of    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 and which encode the impact of new physics coupled the standard SU(2) gauge boson Peskin:1991sw (). Thus the new scalars must also comply with the experimental constraints on these quantities. To ensure this, we employ the pertinent formulas from Ref. Grimus:2008nb () and the and data from Ref. pdg ().

Lastly, the parameters of the scalar potential    in Eq. (8) need to fulfill a  number of theoretical conditions. The quartic couplings in cannot be too big individually, for otherwise the theory will no longer be perturbative. Another requirement is that 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 less than (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  1 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 region evaluated from the observed relic density. We also display the upper limits on inferred from Eq. (5) for the    limit (black dotted curve), from the latest LUX lux () and PandaX-II pandax () searches, and from the aforementioned theoretical demands for perturbativity, potential stability, and unitarity.

The plot in Fig.  5(b) exhibits the corresponding prediction for (green curve) compared to its empirical counterparts obtained from the data depicted in Fig. 1(a) by employing Eq. (20) with    from Set  1 in Table  1. One observes that the   GeV  region, represented by the dotted section of the green curve, is incompatible with the LHC constraint on    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 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.

For further illustrations, in Fig.  6 we graph analogous results with the input numbers from Sets 2 and 3 in Table  1. Their 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 (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 a more straightforward comparison between the model predictions and direct search results, which are typically reported in terms of the DM-nucleon cross-section , we have converted the calculated in Figs.  5 and  6 to the three (green) curves in Fig.  7(a) using Eq. (20) with the values from Table  1 and assuming that the target material in the detector is xenon. Recalling that the DarkSide G2 experiment will employ an argon target dsg2 (), we plot the corresponding predictions for assuming an argon target instead in Fig.  7(b), which reveals some visible differences from Fig.  7(a) in the predictions with  ,  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 -portal THDM II+D parameter space. We notice specifically that the predicted in much of the   GeV  region is under the neutrino-background floor.

### iii.2 λh=0

In this scenario, the cross section of the darkon annihilation into THDM particles is

 σann=σ(DD→H∗→X\textscsm)+\raisebox−7.0pt$\lx@stackrel∑\scriptsizes1s2$σ(DD→s1s2), (29)

where

 σ(DD→H∗→X\textscsm)=4λ2Hv2(m2H−s)\raisebox1.0pt$2$+Γ2Hm2H ∑iΓ(~H→Xi,\textscsm)√s−4m2D, (30)

with being a virtual having the same couplings as the physical and an invariant mass equal to , and the sum in is again over  .  For the -mediated darkon-proton scattering,  ,  the cross section is

 σpel=λ2Hg2pp