Complex Scalar Dark Matter in G 2 HDM

The complex scalar dark matter (DM) candidate in the gauged two Higgs doublet model (G2HDM), stabilized by a peculiar hidden parity (h-parity), is studied in detail. We explore the parameter space for the DM candidate by taking into account the most recent DM constraints from various experiments, in particular, the PLANCK relic density measurement and the current DM direct detection limit from XENON1T. We separate our analysis in three possible compositions for the mixing of the complex scalar. We first constrain our parameter space with the vacuum stability and perturbative unitarity conditions for the scalar potential, LHC Higgs measurements, plus Drell-Yan and electroweak precision test constraints on the gauge sector. We find that DM dominated by composition of the inert doublet scalar is completely excluded by further combining the previous constraints with both the latest results from PLANCK and XENON1T. We also demonstrate that the remaining parameter space with two other DM compositions can be further tested by indirect detection like the future CTA gamma-ray telescope. 1 ar X iv :1 91 0. 13 13 8v 1 [ he pph ] 2 9 O ct 2 01 9


I. INTRODUCTION
Dark Matter (DM) has become one of the most discussed topics in cosmology, astrophysics and particle physics. However, besides the indirect evidence from the power spectrum from the cosmic microwave background radiation (CMB) and the galaxy rotational curves which provide strong hints for the need of DM, current experiments for DM direct detection, indirect detection and collider searches still show no clue for the nature of DM. Currently, the best description of the early history of the Universe is given by the ΛCDM model which assumes the presence of dark energy and cold DM in additional to the ordinary matter. The leading hypothesis is that this cold DM is comprised of weakly interacting massive particle (WIMP) that was thermally produced just like the other Standard Model (SM) particles in the early universe. It is well known that the most compelling feature of WIMP DM is that, after freeze-out whence the DM reaction rates fell behind the Hubble expansion rate of the universe, it is possible to achieve the correct relic abundance with an electroweak sized annihilation cross section with a WIMP mass of a few hundreds GeV to a few TeV.
On the collider phenomenology side, we now know that all the major decay modes of the SM 125 GeV Higgs discovered on the fourth of July 2012 at the Large Hadron Collider (LHC) have been observed except the Zγ and µ + µ − modes. So far all the experimental results agree with the SM predictions within 10 ∼ 15%. Nevertheless there are still some rooms for new physics. A particular class of models that extends simply the scalar sector of the SM to address new physics is quite popular. The most well known example is the general two Higgs doublet model (2HDM), which has several variants and resulted in very rich phenomenology. For a review of 2HDM and its phenomenology, see for example [1,2]. One of the interesting variants of 2HDM is to impose a Z 2 symmetry in the model so that the second Higgs doublet is Z 2 -odd and then can be a DM candidate. This is the inert Higgs doublet model (IHDM) [3] and many detailed phenomenological studies  had been performed over the years. Furthermore, the idea that this Z 2 symmetry emerges accidentally in a renormalizable gauged two Higgs doublet model (G2HDM) has been explored recently in [28]. In G2HDM the two Higgs doublets are grouped together in an irreducible doublet representation of an extra non-Abelian SU (2) H gauge group.
Besides the new hidden SU (2) H , the SM gauge group is extended by including a new U (1) X symmetry. The scalar sector is further extended by including a new SU (2) H doublet and a triplet, both singlets under the SM gauge group.
Although any electrically neutral Z 2 -odd particle in G2HDM can be considered a DM candidate, such as W or heavy new neutrinos, in this work we choose to concentrate on the phenomenology of the complex scalar DM candidate. The reason is mainly that we want to present this model as a viable alternative to IHDM and as such our setup is focused in providing a light, neutral and Z 2 -odd complex scalar.
Another reason is practicality, given that our setup is complicated enough to distinguish at least 3 main types of DM candidates coming only from the scalar sector due to mixing effects. We will study in detail the differences, similarities and results of these three possibilities. Phenomenologically, we expect all of them to communicate with the SM through Higgs portal type interactions [29,30]. However, we will demonstrate that the SM Z boson as well as its heavier siblings in G2HDM will also play non-negligible roles as mediators in various DM processes in relic density, direct and indirect detection, especially for the inert doublet-like DM case.
In the past, some collider phenomenology of G2HDM has been studied [31][32][33]. It was determined that Drell-Yan type signals may help detect the G2HDM Z in the high luminosity upgrade of the LHC [32] and that enhancement of pair production of Higgs boson in the LHC is moderate compared to the SM [33]. In a recent study it has been determined that G2HDM has a viable scalar sector parameter space [34], compatible with vacuum stability and perturbative unitarity conditions, as well as Higgs phenomenology constraints from the LHC. The gauge sector is constrained by electroweak precision tests (EWPT) [35], setting limits on the masses of the new gauge bosons and the gauge sector parameter space. It is precisely this two recent studies on the scalar and gauge sectors constraints (SGSC) that we will take as starting point for our study, thus ensuring that the final constrained parameter space is consistent with previous studies and that our result has a stronger relevance.
This article is organized as follows. In Sec. II we briefly recall some salient features of the G2HDM model, in particular the scalar potential and mass spectra. In Sec. III we point out after spontaneous symmetry breaking there exists an accidental discrete Z 2 symmetry in the whole Lagrangian of G2HDM. We classify all the particles in the model according to whether they are even or odd under this discrete symmetry, dubbed as h-parity. This residual symmetry forbids the lightest particle in the hidden sector to decay and hence it, if electrically neutral, may be a cold DM candidate in the model. We discuss further the compositions of complex scalar dark matter that is relevant in this work. In Sec. IV, we discuss the DM constraints included in our analysis and how they are affected by our setup in more general terms. We describe how relic density (RD), direct detection (DD) and indirect detection (ID) measurements constrain each of the three different compositions of complex scalar DM considered. We also discuss the collider searches of DM from the mono-jet plus missing energy search and invisible Higgs decay. In Sec. V, after a brief description of the methodology used in our numerical analysis we present the results of our analysis. Finally, in Sec. VI we summarize our findings and conclude, including a brief comment on future detectability. Some Feynman rules that are most relevant to the processes discussed in this work are collected in the Appendix A.   [42][43][44][45][46][47]. The matter contents of the G2HDM model and their respective quantum numbers are listed in Table I.

Scalar Potential
For this work we will be using the scalar potential from Ref. [34], that extends the original potential of Ref. [28] by adding two new terms with couplings λ H and λ HΦ . The most general scalar potential that respects the G2HDM symmetries can where the parameters ξ and η can have any value in the ranges 0 ≤ ξ ≤ 1 and On the other hand we have to make sure that our parameter space remains within perturbative limits. Again we look only at quartic couplings since 2→2 scattering processes induced by cubic couplings are suppressed by their propagators while quartic couplings are not. After checking all the possible 2→2 scattering processes the final ranges allowed by perturbative unitarity are where with λ + H ≡ λ H + λ H /2 and λ 8,9,10 given by the three roots of the equation On the phenomenological side, we will require the presence of a SM Higgs with a mass of 125.09 ± 0.24 GeV and a signal strength for the Higgs decay into two photos of µ γγ ggH = 0.81 +0.19 −0.18 as found by the ATLAS experiment [51]. For more details about the theoretical conditions described here, we encourage the interested reader to consult Ref. [34].

Higgs-like (Z 2 -even) Scalars
Expanding the scalar potential in terms of the VEVs and taking the second derivatives with respect to the scalar fields, one can obtain the mass terms and the mixing terms of the scalar fields. The SM Higgs is extracted from the mixing of three real scalars h, φ 2 and δ 3 1 . The mixing matrix of these Z 2 -even neutral real scalars written 1 We follow the notations of [28] shifting the scalar fields as: 12 in the basis of S = {h, φ 2 , δ 3 } T is given by The physical fields with definite mass can be obtained by doing the similarity transformation to this mixing matrix via orthogonal rotation matrix, O, in such a way where the masses of the fields are arranged in ascending manner The interaction basis S and mass eigenstates are related through the O mixing matrix In this setup, the 125 GeV Higgs boson observed at the LHC is identified by the lightest mass eigenstate h 1 .
Other Z 2 -even scalars are the massless would be Goldstone bosons G ±,0 and G 0 H which do not mix with other scalar fields. However they mix with the longitudinal components of the gauge fields and will be absorbed away.

Dark (Z 2 -odd) Scalars
The charged Higgs is sitting at the upper component of H 2 which acquires mass from all three VEVs but it does not mix with other fields. Since H 2 couples to all This matrix has zero determinant, which means that at least one of the mass eigenstates is massless. Despite complex fields, the mass matrix in Eq. (21) is real and symmetric, we can rotate this matrix into its diagonal form through a similarity transformation with the orthogonal matrix O D , The relation between interaction and mass states is given by The first zero eigenvalue in Eq. (22) corresponds to G p,m , the would-be Goldstone boson to be absorbed by W (p,m) , the complex gauge bosons of SU (2) H . Here we assume the hierarchy m 2 D < m 2 ∆ . Note that we strictly avoid degenerate masses to simplify the analysis when D is the dark matter candidate. However, from the mass expressions given below, one can see that very specific parameter choices are necessary to make the two massive states degenerate. The masses of the two physical massive eigenstates are given by where 2 See previous footnote for the definitions of these complex scalars.
14 The lightest state between H ± and D, if lighter than every other Z 2 -odd states, has the possibility to become the DM candidate. However, an electrically charged DM candidate such as H ± is undesirable. For this reason, we will concentrate on parameter space where m H ± > m D .

Gauge Bosons
After SSB, the gauge bosons that acquire mass terms are the B, X, and all the components of W and W . The charged W ± gauge bosons remains completely SMlike with its mass given by M W = gv/2. The W p = (W m ) * does not mix with the SM W ± and acquires a mass given by The remaining gauge bosons, B, W 3 , W 3 and X have mixing terms. We can write their mass terms as a 4×4 matrix using the basis V = {B, W 3 , W 3 , X} T , where M X and M Y are the two Stueckelberg mass parameters [52][53][54][55][56][57][58][59][60] introduced for U (1) X and U (1) Y respectively. This mass matrix has zero-determinant, meaning that there is at least one massless state that can be identified with the photon. The remaining three states are massive in general. One of them, the Z, is related to the SM gauge boson Z SM , and the other two are the extra gauge bosons Z and Z . As in the neutral scalars case, we can diagonalize this mass matrix by an orthogonal We will also use Z i with i = 1, 2, 3 for Z, Z , Z respectively in the following. As noted in Ref. [35], where M Z SM = g 2 + g 2 v/2 is the SM gauge boson Z SM mass. Given the form of the  (28). In that case we can relate the mass eigenstates with the intermediate states as V Z = O Z ·{A, Z, Z , Z } T . Hereafter, we will call O G to the non-diagonal 3 × 3 part of O Z , such that O Z j+1,k+1 = O G j,k with j and k = 1, 2, 3, as explicitly shown in Eq. (6) of Ref. [35]. Note that the photon A remains the same between the intermediate states V Z and the mass eigenstates.
This necessarily means that the only non-zero element in the first column and row of O Z is O Z 1,1 = 1. Interestingly, the only gauge boson that acquires mass contributions from the three non-zero VEVs is the W (p,m) with its mass given in Eq. (25).

MATTER CANDIDATE
As mentioned in the previous session, the stability of the scalar dark matter candidate in this model is protected by the accidental discrete Z 2 symmetry in the scalar potential which is automatically implied by the SU (2) L × U (1) Y × SU (2) H × U (1) X gauge symmetry. Due to its special vacuum alignment where the H 2 field does not acquire a VEV, the accidental Z 2 symmetry remains intact after SSB. It was argued in [28] that there is no gauge invariant higher dimensional operator that one can write down which can lead to the decay of DM candidate in G2HDM. The presence of the accidental discrete Z 2 symmetry after SSB reinforces such argument.
This discrete Z 2 symmetry in G2HDM that we observe here is kind of peculiar in the sense that different components of the SU (2) H doublets H and Φ H , and triplet ∆ H have opposite parity. Thus for dark matter physics it is mandatory to give VEVs to those scalars with even parity. Otherwise the Z 2 symmetry will be broken spontaneously which will lead to no stable DM as well as the domain wall problem in the early universe. Another peculiar feature of this Z 2 symmetry is that it acts on the complex fields. We will refer this accidental discrete Z 2 symmetry as the hidden parity (h-parity) in G2HDM in what follows.  Table II. Thus besides the two well-known accidental global symmetries of baryon number and lepton number inherited from the SM, there is also an accidental discrete Z 2 symmetry in G2HDM. Other than protecting the stability of the lightest electrically neutral Z 2 -odd particle to give rise a DM candidate, this accidental Z 2 symmetry also provides natural flavor conservation laws for neutral currents [61,62] at the tree level for the SM sector in G2HDM [28], as described in previous paragraph. While it is important to unravel if the h-parity in G2HDM has a deeper origin from a larger theoretical structure, for example like grand unification or supersymmetry or braneworld, we will not pursue further here.
In principle, any electrically neutral Z 2 -odd neutral particle can be a DM candidate (e.g. the heavy neutrinos ν H , the complex scalar mass eigenstate D and the gauge boson W (p,m) ). In this work, we focus on the lightest Z 2 -odd complex scalar field D. From Eq. (21) we know that D is a linear combination of the interaction where O D ij represents the (i, j) element of the orthogonal matrix O D . The actual values of the elements of this matrix depend on the actual numerical values of the parameters in Eq. (21).

Since a particular dominant component cannot be inferred from Eq. (21) together
with the constraints presented in Sec. II B, we take the approach of considering three different main compositions. Using the rotation matrix elements we can define the Our results will be classified in three different cases: To avoid cluttering in the following, we will use the more concise terms doubletlike, triplet-like and Goldstone-like DM to refer to the above cases of 1, 2 and 3 respectively.
Correlation between the ratio v ∆ /v Φ and the composition mixing parameter f G p for all the DM types after applying constraints from the scalar and gauge sectors.
In order to realize any one of the three cases of the DM discussed above, one needs to have its diagonal element in the mass matrix given by Eq. (21) to be the lightest, while its mixings with the other two off-diagonal elements are small. However, the mixing among the other two can be arbitrary. Take the Goldstone-like DM as an example. It is easy to note that the (1,1) and (3,3) elements of the mass matrix in Eq. (21) have a see-saw behaviour controlled by the value of v ∆ . The (2,2) element remains almost unaffected thanks to the term proportional to large v 2 Φ . Goldstone-like DM is characterized by a large value in the (1,1) element of Eq. (21) when compared to the (1,2) and (1,3) elements, given by λ HΦ vv Φ /2 and −M Φ∆ v Φ /2 respectively, so as to suppress the mixing effects. The size of the (1,2) element is not relevant since it is proportional to the smaller term vv Φ as compared with both the (1,1) and (2,2) elements which are always much larger. The difference in size between the (1,1) and (1,3) elements is best measured by taking the ratio between them which is roughly about 2v ∆ /v Φ . In other words, the v ∆ /v Φ ratio controls the Goldstone boson composition of the DM mass eigenstate. This is illustrated in Fig. 1, where the correlation between the ratio v ∆ /v Φ and the composition mixing parameter f G p is shown for all DM types. The small arc in the correlation curve with f G p > 2/3 is highlighted by red color indicating only a small parameter space is allowed for Goldstone-like DM. Note that when the ratio v ∆ /v Φ grows close to 1, In this section we briefly describe each of these experimental constraints used in this analysis.

A. Relic Density
It is fascinating to wonder about the thermal history of DM based on all our current knowledge of physics. The simplest scenario is that a WIMP maintains its thermal equilibrium with the SM sector before freeze-out and the DM number density can be described by a Boltzmann distribution. Therefore, the DM mass determines its abundance before freeze-out. As in most WIMP theories, owing to the small DM-SM couplings, the relic density comes out too large and the correct abundance can be only achieved by some specific mechanisms. The mechanisms to reduce the thermal DM relic density in the G2HDM can be both from DM annihilation and also from coannihilation with heavier Z 2 -odd particles. Coannihilation only happens if the next lightest Z 2 -odd particles are slightly heavier than DM (usually 10%) so that its number density at the temperature higher than freeze-out does not suffer a large Boltzmann suppression. In our setup, the heavier Z 2 -odd scalar ∆, the charged Higgs, new heavy fermions, or gauge boson W (p,m) can coannihilate with the DM candidate D. Additionally, the SM Higgs and Z resonance can play an important role for the doublet-like DM while there is no Z resonance in the triplet-like and Goldstone-like DM cases because both ∆ H and Φ H are SM singlets. As we will see later, the couplings between DM and some of the mediators in G2HDM could be suppressed by mixings or cancellations.
The scalar ∆ and the DM candidate D come from the same mass matrix. The splitting between their masses is mostly controlled by the second term in the nu- For the doublet-like DM case, the mass of the DM candidate is close to the mass 22 of the charged Higgs with the splitting approximately given by in the approximation where DM mass is dominated by the (2,2)  To compare against experimental data, we will consider the latest result from the PLANCK collaboration [63] for the relic density, Ωh 2 = 0.120 ± 0.001. In particular, we will require the parameter space of G2HDM to reproduce this well measured value with a 2σ significance.

B. Direct Detection
The most recent constraint for DM direct search is given by the XENON1T collaboration [64]. The null signal result from this search puts the most stringent limit on DM nucleon cross section so far, especially for the DM mass that lies between 10 GeV to 100 GeV. The XENON1T collaboration excluded DM-nucleon elastic cross sections above 10 −46 cm 2 for a DM particle with mass around 25 GeV.
In models with isospin violation (ISV), DM interactions with proton and neutron can be different and the ratio between the DM-neutron and DM-proton effective couplings, f n /f p , can have values that differ from 1 significantly depending on the model parameters. In particular, for a target made of xenon, the ratio f n /f p ≈ −0.7 corresponds to maximal cancellation between proton and neutron contributions [65].
For instance, if DM interacts with nucleons mediated by the Z boson, the strength is characterized by the electric charge and the third generator T 3 of SU (2) L group.
The vectorial coupling of quark q (u or d-type) to the SM Z boson in G2HDM is Due to different Q q , T 3 , T 3 and X charges, this coupling is expected to vary depending on the quark q being u or d-type.
where N stands for a nucleus with mass number A and proton number Z. For definiteness, we will ignore all the isotopes of xenon and fix A and Z to 131 and 54 respectively in this work. We obtain the effective couplings f p and f n by using micrOMEGAs [66]. The DM-nucleon reduced mass is denoted as On the other hand, the limit published by XENON1T is for the nucleon with isospin conserving assumption f n = f p . To reconstruct the XENON1T results at the 24 nucleus level for general value of the ratio f n /f p , we use the following expression where µ 2 p is the DM-proton reduced mass. In this work, we use Eq. (33) to constrain our direct detection prediction. Since we are dealing with complex scalar DM, we need to consider the antiDM interaction with the nucleon. The DM-nucleon interaction and antiDM-nucleon interaction in general can be quite different. When the mediators are heavy enough, one can integrate them out to obtain effective interactions for the DM and nucleon.
The spin independent interaction for complex scalar DM can be written in terms of effective operator as [67] where the ψ N , λ N,e , and λ N,o denote the nucleon field operator, the coupling of even operator, and the coupling of odd operator respectively. The effective coupling of DM (antiDM) with the nucleon is given by where the plus (minus) sign stands for DM-nucleon (antiDM-nucleon) interaction.
The first term in the right hand side of Eq. (34) represents the even operator interaction between DM and the nucleon. It is called even operator because when one exchanges D with D * , the interaction stays the same. On the other hand, under a similar exchange between D and D * the second term flips sign. Thus, it is called odd operator. As a result, the interaction strength between DM-nucleon and antiDMnucleon will not be the same and it is given by Eq. (35). Hence the numerical value of σ D * N is in general not equal to σ DN given by Eq. (32) because the effective couplings Excluding the early universe, DM at the present may also annihilate to SM particles significantly at the halo center where DM density is dense enough to produce cosmic rays or photons which can be distinguished from those standard astrophysical background. Such a measurement is known as DM indirect detection. As long as indirect detection constraints are concerned, the continuum gamma-ray observations from dwarf spheroidal galaxies (dSphs) can usually place a robust and severe limit on the DM annihilation cross section for DM masses larger than 10 GeV [68]. This is owing to two advantages of searching DM at the dSphs. First, the dSphs provides an almost background-free system because they are faint but widely believed to be DM dominated systems. Second, their kinematics can be precisely measured, hence the systematical uncertainties from DM halo can be controlled. Therefore, in this work we will only use the dSphs constraints implemented in LikeDM The standard gamma-ray fluxes produced from DM annihilation at the dSphs halo is given by where J = dldΩρ(l) 2 is the so-called J-factor, which integrates along the line-of-

D. Collider Search
Mono-jet Search DM particles could be produced copiously at colliders. Unfortunately, DM can not be detected on its own since it would pass through detectors without leaving any trace. Therefore, one should look for the DM production associated with visible SM particles. At the LHC, the signal of an energetic jet from initial state radiation that balances the momentum of undetected DM, usually referred to mono-jet signal, is one of the sensitive channels to the search for DM. As shown in Fig. 3, the DM pairs are mainly produced in the Feynman diagrams with the exchanges of Z 2 -even Higgs bosons and neutral gauge bosons in the G2HDM. For numerical study, we take the parameters allowed by EWPT [35] and the XENON1T constraints (to be discussed in Sec. V), and find out that the cross sections are far below the current limits set by ATLAS [71] and CMS [72] collaborations at the LHC. Therefore the mono-jet search would not play any significant role in determining the viable parameter space for DM in G2HDM.

Invisible Higgs Decay
The Higgs boson will decay into a pair of DM when the DM is lighter than half its mass. This decay channel is known as the invisible decay of the Higgs boson.
At tree level in G2HDM, the partial decay width of the Higgs boson to pair of dark matter, h 1 → DD * , is given by where the λ DD * h 1 coupling depends on the composition of the h 1 . For example, for the triplet-like DM case, it can be deduced from Eq. (A6), viz., Currently the upper limit on the Higgs invisible decay branching ratio is rather loose, about 24% at 95% C.L. [74] at the LHC. Taking m D m h 1 together with SM Higgs total decay width of 13 MeV [74], the LHC limit implies an upper bound, However, we found this limit is not as stringent as DM direct detection unless m D 10 GeV where DM recoil energy is below the XENON1T threshold.

A. Methodology
In order to keep consistency with previous G2HDM studies, in particular the scalar sector constrains presented in [34], we will perform random scans to generate a sample of points consistent with all the conditions mentioned there. In our case, we will not keep v Φ fixed. Due to Z search constraints [35], we start our scan range at v Φ = 20 TeV. Considering the energy scale for future colliders, we scan v Φ up to 100 TeV 3 .
We will complete the scan with the free parameters of the gauge sector g H and g X , while fixing the Stueckelberg mass parameter M X = 2 TeV corresponds to the heavy M X scenario discussed in [35]. We will keep the g H coupling below 0.1 to avoid the Drell-Yan constraints. The lower bound of g H will be decided point by point such that the W boson is heavier than the DM D. From Eq. (25), we can obtain a condition for the minimum value of g H Additionally, we will require that the gauge bosons Z and Z are both heavier than the SM-like Z and that the latter has a mass within its 3σ measured value of 91.1876 ± 0.0021 GeV.
To keep heavy fermions above detection limits, we will consider their masses to be no less than 1.
to be reasonably small in order to minimize their effects on perturbative unitarity and renormalization group running effects. Therefore, we use the following formula to determine the appropriate Yukawa couplings for each point in our scan Given the size of v Φ and the fact that m D has to be the lightest Z 2 -odd particle, we expect that Eq. (41) to easily remain below 1 for all our parameter space. Thus, in this set up, one expects most coannihilation contributions are coming from other Z 2 -odd particles such as∆, H ± and W .
From these two steps we collect ∼ 5 million points that include numerical values for model parameters, and results from scalar and gauge bosons masses, and 4 We note that while the Yukawa couplings among the SM fermions and the neutral Higgses Applying this condition increases the abundance of solutions where the lightest complex scalar composition is dominated by H 0 * 2 . This explains the far more limited scan range for the parameter λ HΦ for the doublet-like DM case. The complete set of parameters scanned and their ranges can be found in Table III. Note that in Table III, the different ranges for M H∆ , M Φ∆ , v ∆ and v Φ are selected for the three cases so that we can easily find the corresponding DM composition.
In particular, the very different and smaller fine-tuned ranges of v ∆ and v Φ in the Goldstone-like column are due to this composition being present for v ∆ /v Φ ≈ 0.8 but limited by EWPT to be less than ∼ 0.9, as demonstrated earlier near the end of Sec. III.
Before embarking upon the numerical results, we make some comments on the Sommerfeld enhancement [75,76]    quite massive and not too distinct from each other, we do not expect significant Sommerfeld enhancement in G2HDM. Certainly a more decent study is necessary in order to provide a definite answer. Furthermore we will see in our analysis below that the direct detection limit from XENON1T will provide more stringent constraints than the current indirect detection results from Fermi-LAT. We will ignore such effects in the present analysis.

B. Results
To ease the discussion of our numerical results, it is useful to divide the DM mass range into several regions:  In the following, we discuss in more detail the DM annihilations for this inert doublet-like DM case in the four DM mass regions (i) to (iv) consecutively.
(i) First, the DM masses that lies between 1 GeV to 10 GeV whose major contributions of the DM annihilation cross section are given by DD * → cc and τ + τ − via s-channel SM Higgs exchange. Despite of the small c and τ Yukawa couplings, the cross section can be slightly enhanced by the relatively big DD * h i coupling, as given in Eq. (A4). Thanks to large values for λ HΦ v φ and λ H∆ v ∆ . diagrams. The sum of these 3 diagrams is proportional to (s + t + u − 2m 2 D − 2m 2 Z ) and hence vanishes identically due to kinematical constraint. Thus the remaining diagrams for DD * → Z L Z L are given by the s-channel h i exchange which lead to S-wave total cross section in the non-relativistic limit. There is a similar cancellation between the 4-point contact interaction diagram and the t-channel charged Higgs exchange diagram for the W + L W − L final state. The sum of the amplitudes from these two contributions is given by where t = m 2 D + m 2 W − s/2. Clearly, when s is sufficiently large such that all masses can be ignored and t ∼ −s/2, the above amplitude vanishes. However one notes that if D−H ± coannihilation happens for this heavy DM mass region, i.e., when m D m H ± , the above amplitude is also vanishing. Thus in the heavy DM mass region where the D − H ± coannihilation occurs, the dominant diagrams that contribute to DD * → W + L W − L are the h i and Z i exchanges which give rise to S-wave and P -wave total cross sections respectively in the non-relativistic limit. We can also conclude that the total cross sections for DM annihilation into both W + L W − L and Z L Z L final states in G2HDM are consistent with unitarity [79].
In the right panel of Fig. 4 predicts a typical value of the cross section of order 10 −38 cm 2 . It can be excluded by XENON1T [64] and CRESST-III [80] down to DM masses above 2 GeV. For the points below 2 GeV that survive the CRESST-III constraint, the predicted relic abundance is always higher than the measured PLANCK value. Regarding the ISV effects, we check that |f n /f p | remains typically 3 orders of magnitude far away from the maximal cancellation value of f n /f p ≈ −0.7. Therefore, there is no noticeable reduction in the nucleon-level DD cross section.
It is clear from the previous discussion that for the doublet-like DM case there is no surviving parameter space that can remain after the constraints from both PLANCK and XENON1T are taken into account. Therefore, doublet-like DM in G2HDM is completely ruled out by current experiments, at least under the somewhat generic conditions set up in this paper. A study of particular mechanisms or very specific sets of parameters (e.g. a very light mediator region) that may bring down the relic density for light doublet-like DM (∼ 1 GeV) while keeping the prediction of direct detection intact is out of the scope of the present analysis.

SU (2) H Triplet-like DM
One fundamental difference between triplet-like and doublet-like DM is that now D is dominated by the term O D 32 ∆ p in Eq. (29). Therefore, one should expect all the couplings to behave differently from the previous doublet-like case. In particular, the coupling terms that were relevant for doublet-like DM will now be suppressed The gray area in the right panel is excluded by PLANCK data at 2σ. In the right panel, the lower red solid line is the published XENON1T limit with isospin conservation, while the upper green solid line is the same limit but for ISV with f n /f p = −0.5. Some orange filled squares are above the published XENON1T limit due to ISV cancellation at nucleus level. In region (iii) where m h 1 /2 < m D < 500 GeV, the relic density reduction mechanism is similar to the doublet-like DM case discussed above. The annihilation cross section is highly dominated by W + W − (more than ∼ 50%), h 1 h 1 (∼ 25%), and ZZ The contribution from h 3 exchange is negligible because of its heavy mass.
In region (iv) where m D > 500 GeV, DM annihilates into W + L W − L predominantly while other channels are subdominant, similar to the doublet-like DM case. There is no need to elaborate further here.
Generally speaking, the charged Higgs H ± contribution here can be omitted since it is more than twice heavier than the DM D. Differently from the doublet-like case, there is no coannihilation between H ± and D in the triplet-like DM case.
Next, the coannihilation between DM and ∆ is absent as well because the ∆ is also much heavier than D due to the choice of larger v ∆ to make the (3,3) entry of Eq. (21) smaller. Therefore, the only possible efficient coannihilation is between DM and W for DM mass above 400 GeV (orange boxes at the left panel of Fig. 5).
This coannihilation is only important for relic density above 0.12, where some DD * annihilation channels may be insufficient because their couplings to h i and Z j may be suppressed. A small region with heavy fermion coannihilation happens for m D > 1 TeV with relic density above 10 (green shaded points in the left panel of Fig. 5). This is close to the maximal relic density in our scan for that mass range. This indicates that heavy fermion coannihilation is important only when the other annihilation channels are strongly suppressed.
Regarding direct detection, due to the DD * Z coupling suppression by mixings in In the right panel of Fig. 5, we can see that for m D 300 GeV it is possible to find a region that agrees with relic density constraint from PLANCK at 2σ and remains below the published XENON1T limit at the neutron with f n /f p = 1. Note that some of the allowed points (orange squares) are above this XENON1T limit.
This is due to mild ISV cancellation that brings such points below the XENON1T limit at nucleus level, as given by Eq. (33). For comparison, the XENON1T limit at the neutron level with ISV of f n /f p = −0.5 is also shown.
The constraint of indirect detection from Fermi-LAT's gamma-ray observation imposed on the triplet-like DM is shown in Fig. 6. The left panel presents the DM annihilation cross section dependence 5 on the DM mass at the present universe with SGSC+RD. Results are only presented for the dominant annihilation channels, bb and W + W − . One can see that DM with m D 90 GeV mainly annihilates to bb.  limit with isospin conservation, while the upper green solid line is the same limit but for ISV with f n /f p = −0.5. Some blue filled squares are above the published XENON1T limit due to ISV cancellation at nucleus level.
At the region near the Z or h 1 resonance, the corresponding cross section at the present universe drops while satisfying the relic density. This is a typical feature of the resonance region because the DM relative velocity at the early universe is much larger than the value at the present one. In order to cancel a large cross section caused by the resonance at the early universe, a small coupling of DD * Z or DD * h 1 is required to make σv at the early universe comes close to the canonical value of 42 10 −26 cm 3 ·s −1 . However, when the universe temperature drops, the resonance cannot be maintained by the kinetic energy of DM at the present day. At this time the cross section becomes smaller and is hard to be observed by Fermi-LAT.
Once DM mass is heavier than W ± boson mass, the final state W + W − starts dominating the annihilation cross section rapidly. Note that the current ID sensitivity can only apply strongly for the DM mass located between 10 GeV and few hundred GeV. However, the future CTA sensitivity [81] might reach the TeV region of m D and further constrain our parameter space, as show in the left panel of Fig. 6.
In the right panel of Fig. 6, we display the exclusion from ID projected on the plane of DM-neutron spin independent cross section σ SI n versus m D . We can see that all the ID excluded points sit above the limit set by XENON1T. The exclusion limits are given by recent XENON1T data (blue unfilled squares) and Fermi gamma-ray constraints (orange crosses). One can see the XENON1T exclusion power is much stronger than Fermi gamma-ray exclusion.  panel of Fig. 8 shows the zoomed in region of points on the (m D , σ SI n ) plane allowed by the SGSC+RD+ID and SGSC+RD+ID+DD. As mentioned before, ISV effect (f n /f p ≈ −1.86) reduces the sensitivity of the XENON1T result and some points pass all the constraints (SGSC+RD+ID+DD) even though they are above the direct detection limit at nucleon level. Note that there are no points satisfying SGSC+RD+DD+ID beyond m D ∼ 1 TeV in this Goldstone-like case.

C. Constraining Parameter Space in G2HDM
From previous sections, we have learned that the doublet-like DM scenario cannot fulfill the DM constraints and that the Goldstone-like DM requires some fine-tuning in the parameter space and to escape the XENON1T limit a particular value of f n /f p ≈ −1.86 is required. Therefore, we will be focusing on discussing the allowed G2HDM parameter space based on the triplet-like DM.
In Fig. 9, we present the allowed regions of the quartic couplings from the SGSC constraints (green region) and SGSC+RD+DD constraints (red scatter points).
Comparing the green regions with the red scatter points in Fig. 9, one can easily obtain the following results: • The allowed ranges on λ H and λ H remain more or less the same before and after imposing RD+DD constraints. allowed DM mass values range from hundreds of GeV to a few TeV. This range is reflected in g H since the minimal value we choose for g H is given by Eq. (40) and depends directly on the DM mass.
• The other 4 parameters g X , v ∆ , M H∆ and M Φ∆ are not sensitive to the dark matter physics constraints from RD+DD.
In summary, given the setup of the parameter space in our numerical scanning, a good WIMP candidate in G2HDM is the triplet-like complex scalar with a mass m D in the electroweak scale, and it requires g H 2 × 10 −2 and v Φ 30 TeV.

VI. SUMMARY AND CONCLUSION
The G2HDM is a novel two Higgs doublet model with a stable DM candidate protected by an accidental discrete symmetry (h-parity) without the need of imposing it by hand as in the IHDM. After SU (2) H symmetry breaking, the symmetry remains intact and one can find three electrically neutral potential DM candidates with odd h-parity: the lightest dark complex scalar D, heavy neutrino ν H , and the SU (2) H gauge boson W (p,m) . Though these three candidates are all interesting, we focus this paper on the most popular one, the new scalar DM D, which is complex and hence differ from the DM in IHDM. Unlike IHDM, the mixing between Z-odd scalars adds a touch of complexity since DM in G2HDM not only comes from the inert doublet but may also be SU (2) H Goldstone-like and triplet-like. We took the dominant composition (f j > 2/3 with j = H 2 , ∆ p , G p ) as a criteria to classify them but the mixture between them can be simply inferred. In this paper, we have discussed these three types individually with two assumptions: that all the new non-SM heavy fermions are heavy enough to have mostly negligible contributions and that DM were thermally produced before the freeze-out temperature. We have comprehensively shown their detectability and exclusions by the current SGSC and DM constraints (mainly RD+DD).
Because the DM candidate is chosen to be a complex scalar in G2HDM, the DM phenomenology becomes very rich since it has captured both features of the Higgsportal and vector-portal DM models discussed in the literature.
For the inert doublet-like DM, we found some interesting features. First, the main difference between the inert doublet DM in IHDM and G2HDM is that in IHDM there is in general a mass splitting between the scalar S and pseudoscalar P components of H 0 2 , while in G2HDM they are completely degenerate and combined into one single complex field H 0 2 = S + iP . Recall that in IHDM there is only ZSP derivative coupling but no ZSS and ZP P derivative couplings. As long as the mass splitting between S and P remains larger than the exchange energy between DM and nucleons in the direct detection experiments, the interactions mediated by the Z gauge boson are suppressed in IHDM. Since this splitting does not exist in G2HDM, such interactions are unsuppressed and they can bring the spin independent cross section up to ∼ 10 −38 cm 2 , which is significantly above the XENON1T 95% C.L. limit for m D 10 GeV and above CRESST-III result for m D 2 GeV (Fig. 4 right   panel). On the other hand, for m D 10 GeV, the DM is over abundant because of on-shell annihilation channels in cc and τ + τ − (Fig. 4 left panel). Hence, we conclude that the inert doublet-like DM can be completely excluded by SGSC+RD+DD constraints.
Next, a SU (2) H triplet scalar like DM was discussed. Since the composition f H 2 has to be tiny in order to avoid the tension with DM DD, the triplet-like DM can mostly mix with the Goldstone boson G p . There is no Z-resonance region in the triplet-like DM for DM annihilation and the parameter space is more or less consistent with Higgs portal DM. However, DD is still the most stringent constraint comparing with ID and collider constraints. The allowed DM mass by SGSC+RD+DD is required to be heavier than m D 300 GeV (Fig. 5 right panel). Despite weaker constraints coming from ID ( Fig. 6 left panel) and collider searches, it might be possible to detect the heavy DM mass region by the future CTA and 100 TeV colliders even if a DM signal is not found at direct detection experiments before hitting the neutrino floor. As shown by the blue solid boxes in the right panel of Fig. 6, the allowed triplet-like DM mass consistent with SGSC+RD+DD+ID is 300 GeV.
For the last case of the Goldstone-like DM, we found that it is not possible to obtain a pure Goldstone-like DM. The non-tachyonic DM condition and EWPT constraints prohibit the composition f G p > 0.75 (Fig. 1), unless one would like to move to a more fine-tuned region of parameter space. Thus there is a significant component coming from the triplet in the Goldstone-like DM. Because of the Pwave suppression of the Z and Z exchange in the dominated channels of bb and W + W − , the annihilation cross section happens to be smaller than for the triplet-like case and lesser points within the PLANCK relic density measurement ( Fig. 7 left panel). Furthermore, XENON1T measurement excludes almost all the points with appropriate relic density, except for those with a particular value of isospin violation (f n /f p ≈ −1.86) where the sensitivity at XENON1T is reduced. Therefore, only a small region of orange boxes in the right panel of Fig. 7 with m D in the range of 150 ∼ 600 GeV can pass all the SGSC+DD constraints implemented in this work. For ID, the annihilation cross section at the present time for the Goldstone-like DM is typically smaller than the limit from Fermi gamma-ray constraints ( Fig. 8 left panel).
With significant ISV, only the Goldstone-like DM with a mass in the window of 150 ∼ 600 GeV can be consistent with SGSC+RD+DD+ID, as given by the blue solid boxes in the right panel of Fig. 8.
We also presented the impact of DM constraints on the G2HDM parameter space in Figs. 9 and 10 for the triplet-like DM. In this case, we found that the following parameters λ Φ , λ H∆ , λ Φ∆ , g H , and v Φ are significantly constrained by DM constraints, mainly RD+DD, while the four parameters g X , λ ∆ , v ∆ , and M H∆ remains more or less the same as given by the SGSC. It is interesting to note that the SGSC con-straints on g H and v Φ as studied in [34,35] are now further constrained by RD+DD.
We note that the lower limit of g H > 7.09 × 10 −3 for v Φ < 100 TeV is reachable by the future linear (lepton-antilepton) and 100 TeV hadron colliders.
Before closing, we would like to make a few comments. Originally the SU (2) H triplet field ∆ H was introduced to give mass to the charged Higgs (Eq. (20)) in [28] where the two parameters λ H and λ HΦ were missing. With these two extra parameters included, the triplet field ∆ H is no longer mandatory. We note however that the triplet field ∆ H can give rise to a non-singular 't Hooft-Polyakov monopole for the hidden SU (2) H which can play the role as DM as studied in [82] 6 . Nevertheless, one can have a minimal G2HDM without the triplet field. Then the DM D in this minimal model would be just mixture of the inert Higgs H 0 * 2 and the Goldstone field G p H . From the analysis in this work, we know that this DM scenario must be highly fine-tuned in the parameter space due to SGSC+RD+DD. A more interesting alternative DM candidate in this minimal G2HDM is the W (p,m) , which certainly deserves a separate study. Finally, whether the accidental discrete symmetry of h-parity, identified here in the renormalizable Lagrangian for classification of all particles in G2HDM, has a deeper origin remains to be explored in the future. 6 We thank P. Ko for bringing this reference to our attention.