Signals of New Gauge Bosons in Gauged Two Higgs Doublet Model

Recently a gauged two Higgs doublet model, in which the two Higgs doublets are embedded into the fundamental representation of an extra local $SU(2)_H$ group, is constructed. Both the new gauge bosons $Z^\prime$ and $W^{\prime (p,m)}$ are electrically neutral. While $Z^\prime$ can be singly produced at colliders, $W^{\prime (p,m)}$, which is heavier, must be pair produced. We explore the constraints of $Z^\prime$ using the current Drell-Yan type data from the Large Hadron Collider. Anticipating optimistically that $Z^\prime$ can be discovered via the clean Drell-Yan type signals at high luminosity upgrade of the collider, we explore the detectability of extra heavy fermions in the model via the two leptons/jets plus missing transverse energy signals from the exotic decay modes of $Z^\prime$. For the $W^{\prime (p,m)}$ pair production in a future 100 TeV proton-proton collider, we demonstrate certain kinematical distributions for the two/four leptons plus missing energy signals have distinguishable features from the Standard Model background. In addition, comparisons of these kinematical distributions between the gauged two Higgs doublet model and the littlest Higgs model with T-parity, the latter of which can give rise to the same signals with competitive if not larger cross sections, are also presented.

W ′(p,m) , which is heavier, must be pair produced. We explore the constraints of Z ′ using the current Drell-Yan type data from the Large Hadron Collider. Anticipating optimistically that Z ′ can be discovered via the clean Drell-Yan type signals at high luminosity upgrade of the collider, we explore the detectability of extra heavy fermions in the model via the two leptons/jets plus missing transverse energy signals from the exotic decay modes of Z ′ . For the W ′(p,m) pair production in a future 100 TeV proton-proton collider, we demonstrate certain kinematical distributions for the two/four leptons plus missing energy signals have distinguishable features from the Standard Model background. In addition, comparisons of these kinematical distributions between the gauged two Higgs doublet model and the littlest Higgs model with T-parity, the latter of which can give rise to the same signals with competitive if not larger cross sections, are also presented.

I. INTRODUCTION
With the discovery of the 125 GeV scalar boson at the Large Hadron Collider (LHC), the Standard Model (SM) with just one Higgs doublet has now been generally accepted as the standard theory or framework to describe the fundamental strong and electroweak interactions for three generations of elementary particles of quarks and leptons. An extended Higgs sector, however, is often used to address various theoretical puzzles like neutrino masses, dark matter (DM), matter-antimatter asymmetry, hierarchy problem etc. which remain unexplained in this standard framework. Perhaps the general two Higgs doublet model (2HDM), in particular in the context of supersymmetric theories, is the most studied in the literature. Due to its diverse variations, 2HDM has been used as a prototype to address aforementioned theoretical issues. For reviews of 2HDM and its supersymmetric version, see for example [1][2][3]. One of the interesting 2HDM variants is the inert Higgs doublet model [4][5][6][7], in which the neutral component of the second Higgs doublet can be a dark matter candidate due to a discrete Z 2 symmetry imposed on the scalar potential.
Origin of multiple inert Higgs doublets in the context of grand unification has been addressed in [12].
In a recent work [13], we have proposed a novel model, dubbed Gauged Two Higgs Doublet also introduced to render the model free of gauge anomalies. In this model, an inert Higgs doublet can be naturally realized without imposing the ad-hoc Z 2 symmetry mentioned above to accommodate a DM candidate. Flavor changing neutral currents are also absent naturally at tree level. We note that it is widely believed that global symmetry (whether it is discrete or continuous) may be strongly violated by gravitational effects [14,15]. Thus from an effective field theory point of view it is desirable to embed discrete symmetries into local gauge symmetries below the Planck scale [14]. Indeed, besides Z 2 , non-abelian discrete flavor groups like {A 4 , S 4 , A 5 }, {Q 6 , T ′ , O ′ , I ′ }, and {T 7 , ∆ (27), P SL(2, 7)} can be minimally embedded into SO(3), SU (2), and SU(3) respectively [16].
In G2HDM, a distinctive feature is all the SU(2) H gauge bosons Z ′ and W ′ (p,m) are electrically neutral which is not the case for the Left-Right symmetry model (LRSM) [17], the littlest Higgs model with T-parity (LHT) [18], and the original Twin Higgs model (THM) [19].
Naturally one might ask how do we distinguish the G2HDM Z ′ and W ′ (p,m) from other gauge boson impostors which also arise from the non-abelian group SU(2)? In terms of collider searches, some of the new gauge bosons from the aforementioned models can never be singly produced due to the gauge symmetry involved. For instance, the W ′ (p,m) in G2HDM and W ± H in LHT may not be singly produced at the LHC. Generally speaking, the Z ′ and W ′ (p,m) as well as their impostors are short-lived and it is not possible to identify the new gauge bosons using the tracking or displaced vertex techniques designed for long-lived particles.
Hence, in additional to their production cross sections, detailed kinematics distributions have to be involved for making differentiation.
If new gauge bosons can be singly produced, they will be stringently constrained by exotic searches from the LHC [20,21] due to large (resonant) production cross sections. The latest LHC 13 TeV Z ′ resonance searches based on the channels of dilepton [22,23], dijet [20,24], b-quark pair [25], t-quark pair [26], and other bosonic final states [27][28][29] have recently been released. Among these searches, the cleanest dilepton channels yield the most strong constraint on the Z ′ coupling to SM fermions in light of small background. Moreover, from the total electric charge of the decay products it is straightforward to tell singly produced charged bosons from neutral ones.
The current bound from the LHC dilepton searches [22,23] for Z R in the minimal LRSM, [30]. Similarly for Z ′ and W ′ in the Left-Right THM (LRTH) [31] with m W ′ = m Z ′ √ cos 2θ w / cos θ w and g L = g R , one obtains the limit m Z ′ > 3.36 TeV. In some variants of the THM (see e.g. Ref. [32] In this work, we will focus on two benchmark mass spectra (Spectrum-A and Spectrum-B) of the G2HDM for our collider studies. The Spectrum-A contains heavy and decoupled new quarks while the Spectrum-B comprises relatively light new quarks. For all scenarios, new leptons are assumed to be lighter than the additional gauge bosons of interest. Due to the fact Z ′ couples to SM quarks and can be singly produced at the LHC, we first update the bounds on the SU(2) H gauge coupling g H as a function of the Z ′ mass m Z ′ by using the newly released results of the dilepton and dijet searches from the LHC. Next, Z ′ exotic decays into new heavy fermions followed by decays into SM fermions are investigated at the 14 TeV High Luminosity LHC (HL-LHC) and bounds from LHC searches on supersymmetric particles can be applied with simplified assumptions. Then, for the neutral W ′ (p,m) in G2HDM we propose searching for two channels: two leptons and four leptons with missing transverse energy. We shall demonstrate that the pair production of W ′ (p,m) can feature quite distinctive kinematical distributions from the W ′ H pair in LHT, which will be chosen as a representative model for comparisons since W ′ can only be pair produced in both models.
This rest of this paper is laid out as follows. In Sec. II, we briefly review the G2HDM and spell out the relevant gauge interactions for collider searches of interest. In Sec. III, we discuss the methodology employed in the collider simulations. In Sec. IV, we revisit Z ′ direct search limits from the latest 13 TeV LHC data as well as exploring some of its exotic decay channels at the HL-LHC. In Sec. V, signatures for W ′ at a future 100 TeV protonproton collider are scrutinized in detail and compared with those from LHT. We summarize our findings and conclude in Sec. VI. For convenience, we also present the scalar potential of G2HDM and the associated scalar mass spectra in two appendixes. More details of the scalar sector of G2HDM can be found in [33].

II. G2HDM GAUGE INTERACTIONS
In this section, we give a brief review on G2HDM, focusing on gauge interactions that are relevant to our study of collider searches. The particle contents summarized in Table I have the minimal set of new heavy chiral fermions required for anomaly cancellation and new scalars for facilitating spontaneous electroweak symmetry breaking, as proposed in [13].
As mentioned earlier, the two SU ( With a non-vanishing Φ 2 , the four Dirac fields d H , u H , e H and ν H acquire a mass of y ′ d Φ 2 , y ′ u Φ 2 , y ′ e Φ 2 , and y ′ ν Φ 2 , respectively. On the other hand, the SM quarks and leptons obtain their masses from the vev of H 1 via the Yukawa couplings symmetry into an extra U(1) ′ has been studied in [35][36][37][38]. * After symmetry breaking in G2HDM, one can actually show that an effective Z 2 symmetry emerges [34].
There are SU(2) H gauge bosons, W ′(p,m) and W ′3 , and the U(1) X gauge boson X, apart from the SM ones. Due to the symmetry breaking pattern, W ′(p,m) will not mix with the SM counterparts but W ′3 and X mix with the SM SU(2) L W 3 and U(1) Y Y gauge boson.
In this setup, besides the SM massless photon corresponding to the unbroken generator Q = T 3 L + Y , there exists a massless dark photon corresponding to the unbroken generator Q D = 4 cos 2 θ w T 3 generator, and θ w is the Weinberg angle. Such a massless dark photon could be cosmologically problematic. To circumvent the problem, one can resort to the Stueckelberg mechanism to give a mass to the U(1) X gauge boson as in [39]. One could take this mass to be large enough so that X is decoupled from the particle spectra. Another way out is to treat U(1) X as a global symmetry as was proposed in [40] and adopted in [13] as well. We will follow the same strategy in what follows.
In this case, after diagonalizing the mass matrix of Y , W 3 and W ′3 , one obtains massless γ, massive Z and Z ′ . Furthermore, the mixing between Z − Z ′ is constrained to be of order 10 −3 for TeV Z ′ because of the electroweak precision measurements [41]. As a consequence, impacts of the mixing are numerically negligible and will be ignored. The resulting SU(2) H gauge boson mass spectrum is . Note that W ′ (p,m) is always heavier than Z ′ in G2HDM.
As the SM right-handed fermions as well as the new fermions are charged under SU(2) H , they couple to the W ′ (p,m) and Z ′ bosons. The relevant gauge interactions without the Here L(W ) and L(γ) refer to the charged current mediated by the W boson and the electric current by the photon γ respectively, where Q f is the corresponding fermion electric charge in units of e. ∆L represents (electrically) neutral current interactions of the massive bosons, Z, Z ′ and W ′(p,m) (for demonstration, only the lepton sector is shown but it is straightforward to include the quark sector): where The current interactions in L(W ′(p,m) ) and L(Z ′ ) will dictate how W ′ (p,m) and Z ′ decay into SM and heavy fermions, and determine which final states one should look into for collider searches.

III. METHODOLOGY
To simulate the total cross sections and various distributions for the relevant processes in the colliders, we will follow the standard protocol well established by many collider phenomenologists. We use FeynRules [42] to build up the model files for G2HDM and pass it to Madgraph5 [43] for the matrix element calculation and event generation. We simulate parton showering by using Pythia8.1 [44], and employ Delphes3 [45] for detector simulations.
Finally, the package MadAnalysis5 [46] is used to analyze the simulation data.
In the G2HDM, apart from the extra gauge bosons W ′(p,m) and Z ′ , additional heavy fermions have to be included to attain gauge invariant Yukawa couplings as explained above.
To simplify the analysis, we assume two universal masses for the heavy fermions, one for leptons and the other for quarks. As a result, there are five relevant mass scales in our analysis, namely the masses of the dark matter particle H 0 * 2 , the heavy leptons L H = (e H , µ H , τ H ) and ν H = (ν H e , ν H µ , ν H τ ), the heavy quarks Q H = (u H , d H , c H , s H , t H , b H ), and the two heavy gauge bosons W ′(p,m) and Z ′ . In addition, the new charged fermions have to be heavier than 100 GeV, a constraint inferred from the combined analysis of the LEP2 run data by the four LEP collaborations [47]. We will study the following two benchmark mass spectra for the new fermions, while the new gauge bosons are always assumed to be heavier than 1.5 TeV such that the gauge coupling g H is not too small.

Spectrum-A:
Heavy and decoupled new quark scenario.
The new quarks Q H are chosen to be heavier than Z ′ . Specifically we take m Q H = m Z ′ + 1 TeV, and thus channels of the new quarks will not be considered in the Z ′ -resonance searches. On the other hand, the new leptons L H (ν H ) are assumed to be lighter than Z ′ with m L H (ν H ) = 2 m D in which m D is the dark matter mass.
Hence L H (ν H ) can be pair produced by Z ′ on-shell decays. In order to well separate the spectrum, we fix the mass ratio between DM and the SU(2) H gauge bosons: Spectrum-B: Light new quark scenario.
For completeness, we also study a scenario with lighter new quarks where the new heavy quarks and leptons are degenerate: for simplicity decays of the heavy gauge bosons into scalar Higgs pairs are presumed to be either kinematically forbidden or negligible. It is justified since all of the new scalars except for DM can be heavier than W ′ and Z ′ as displayed in the last table in Appendix B.
Moreover, the coupling between the longitudinal components of W ′ and Z ′ and the DM can in principle be made small by varying the parameters in the scalar potential. In this way, the transverse components (whose coupling to DM is simply g H ) govern decays of W ′ and Z ′ into DM particles but this contribution to the decays is subleading compared to those of the heavy fermions in the final states, given the larger number of the new fermions in the model.
In both scenarios, the new heavy fermions are kinematically allowed to be produced by either Z ′ or W ′(p,m) decays. As a result, we propose searches for the new fermions as follows. † • For Spectrum-A, the heavy charged leptons can be produced via pp → Z ′ → L H L H and pp → W ′p W ′m → L H LLL H , and the corresponding final states will be (1) 2l + E T , • For Spectrum-B, the new quark pairs can also be on-shell produced through pp → Z ′ → Q H Q H , and thus the following final states (1) 2j + E T , (2) 2b + E T , and (3) 2t + E T will be considered. These processes are relevant to the dijet plus missing transverse energy searches for Z ′ . Needless to say, the continuum contributions from QCD to the new quark pair production should be taken into account.
Spectrum-B 74.96% 12.52% 12.52% In Table II  fermion pairs, the opening of Q H Q + QQ H final state in Spectrum-B affects only the other exotic leptonic channels. These exotic decays can be phenomenologically interesting as we will see in the next section.
Note that Z ′ and W ′(p,m) can also decay into scalars, H, Φ H and ∆ H , which are charged under SU(2) H . The branching fractions, however, depend on the scalar mixing parameters in the scalar potential [13] which can make the analysis rather convoluted. As mentioned earlier, to simplify the analysis in this work, we neglect scalar final states and instead focus on the fermion channels at which the corresponding partial decay widths are simply fixed by the SU(2) H gauge coupling as well as the new heavy fermion masses.

IV. Z ′ SEARCHES AT THE LHC
In this section, we first present the Z ′ constraints, derived from the ATLAS and CMS dijet and dilepton searches based on the 13 TeV data. Then we propose potential Z ′ signatures from exotic decay searches which have smaller cross sections than direct Z ′ searches but can be explored at the 14 TeV HL-LHC. In Section V, we will investigate the W ′(p,m) searches at a future proton-proton 100 TeV collider. As mentioned before, unlike Z ′ which can be singly created and probed directly by dilepton and dijet searches at the LHC, the heavier W ′(p,m) must be produced in pair in light of the SU(2) H gauge symmetry and therefore are less constrained.
A. Constraints on Z ′ from current dilepton and dijet searches In G2HDM, Z ′ is always lighter than W ′ and can be directly probed by resonance searches as mentioned above. Due to null results of the direct searches, very stringent limits are imposed on any model of Z ′ that directly couples to SM fermions. For instance, the sequential SM with Z ′ having the same couplings to SM fermions as the SM Z gauge boson is constrained to be heavier than 4 TeV [22,23].
Recently, ATLAS and CMS collaborations have reported their updated results of Z ′ resonance searches for channels of dilepton [22,23], dijet [20,24], b-quark pair [25], t-quark pair [26], and other bosonic final states [27][28][29] Table II. mass of the final state particles and yields the most stringent constraints. In this work, we consider two major type of constraints: dilepton and dijet channels. We calculate the cross sections of pp → Z ′ → l + l − /jj with the help of Madgraph5 [43] and compare them with the latest constraints from the LHC.
In Fig. 1, we present the exclusion regions of g H as a function of m Z ′ . The solid lines correspond to Spectrum-A and the dashed lines denote Spectrum-B. The major discrepancies between the two scenarios are the branching ratios of the Z ′ decay into SM quarks and leptons, as shown in Table II. Compared with the previous constraints [13] obtained from the 8 TeV data, the improvement is about a factor of two in the region of 1.5 < m Z ′ < 2.25 TeV. In addition, thanks to the higher center-of-mass energy √ s (8 → 13 TeV), the bounds for m Z ′ > 2.25 TeV are significantly improved and become stronger than the LEP limits based on the σ(e + e − → l + l − ) measurements [13].
We note that the constraints on the Z ′ mass in G2HDM is less stringent than the Z ′ in LRSM or LRTHM where a discrete symmetry is imposed to equate the new gauge coupling to the SM SU(2) L one. The price to pay for G2HDM is of course a smaller g H .

B. Z ′ exotic decays into heavy fermions
We now move on to the Z ′ exotic decays which can shed light on the existence of exotic fermions in G2HDM. As noted before, the scalar decay channels of Z ′ depending on details of the complicated scalar potential and hence are ignored in this work. On the other hand, the heavy fermion channels, which are governed by g H and the heavy fermion masses only, can be easily addressed.
Thinking forwardly and optimistically, one can envisage that a Z ′ will be discovered by the direct searches of dilepton and dijet channels in the foreseeable future at HL-LHC. If so, the heavy fermions in G2HDM can also be probed via Z ′ on-shell decays if kinematically allowed.
In order to perform a more general study for this purpose, we will temporarily relax the slepton (l ± ) searches at the LHC the major process is 2l + E T channel, namely Similarly, in G2HDM decays of Z ′ into a pair of exotic fermions can also lead to the same final states: These two processes with the same final states exhibit analogous event topology, allowing us to apply the same kinematic cuts. § We calculate the cross sections for the Z ′ exotic decays and then impose bounds from SUSY searches on these decays.
The bounds on the heavy fermion masses can be mitigated in a scenario of the compressed mass spectrum: m (L H ,Q H ) m DM . In this case mono-X (X = γ, g, W , Z· · · ) + E T , in particular the mono-jet + E T signal, can be used to search for DM as in the MSSM with the compressed mass spectrum. This scenario, however, will not be considered here as the mass difference m (L H ,Q H ) − m DM is taken to be large.  We now present the constraints on the Z ′ exotic decays. With the modified spectra, we will be able to obtain contours of the production cross sections on the (m Z ′ , m f H ) plane and compare with the LHC limits on the SUSY particle searches.
We concentrate on the following two channels, where l H = (e H , µ H ), and l = (e, µ). In Fig. 2, the contour plots for the cross sections of the processes in (11) and (12)  and m Z ′ − m τ H (right panel). Since the cross section is proportional to g 2 H for onshell heavy fermions ¶ , the results are shown in terms of σ/g 2 H to factor out the g H dependence. For a specific value of g H , one can simply rescale the contours by g 2 H whose limits for a given value of m Z ′ have been presented in Fig. 1. The black, blue, and red contours correspond to σ/g 2 H = 0.1, 0.05, and 0.01 pico-barn (pb) respectively, assuming √ s = 14 TeV.
We employ the recent results of ATLAS SUSY searches for neutralinos and charginos based on the 2l + E T and 2τ + E T channels to constrain the Z ′ exotic decays in G2HDM. The resulting bounds should be regarded as estimated constraints, as the signal regions and efficiency may have some differences between MSSM and G2HDM.
For the left panel of Fig. 2 (2l + E T channel), we use the signal region SR2l-A which refers to a set of event selections listed in Table 1 of Ref. [48]. It gives rise to the constraint ǫσ 95 obs ≤ 1.89 fb at 13 TeV. Assuming that the factor of signal efficiency ǫ is of O(1) at 14 TeV * * , we can infer limits on g H at the 14 TeV LHC in the following way. For instance, to satisfy the SR2l-A bound σ < 1.89 fb, along the black, blue and red contours in the left panel of Fig. 2, the corresponding g H is required to be smaller than 0.137, 0.194 and 0.435, respectively.
Likewise, for the 2τ + E T channel on the right panel of Fig. 2, we utilize the SRC1C1 signal region in Ref. [49]. It yields ǫσ 95 obs ≤ 0.33 fb at 13 TeV that demands g H to be less than 0.057, 0.081, and 0.182 for the black, blue, and red contours, respectively.
Two comments are in order here, regarding the discrepancies between collider searches discussed here for MSSM and G2HDM.
1. The signals of 2l + E T and 2τ + E T in Spectrum-A ′ mainly come from Drell-Yan processes for both G2HDM and MSSM. The major difference between the two models is the distribution of the invariant mass of the final charged leptons. In G2HDM, the invariant mass distributions have a cut-off at m ′ Z , i.e., ¶ For most of the regions of interest, the extra heavy fermions from Z ′ decays are on-shell. * * If the magnitudes of the cross sections at 14 TeV are just slightly larger than those at 13 TeV, the constraints we present here will be more stringent than the 13 TeV ones since the detection efficiency of order 1 has been assumed.  m l + l − , m τ + τ − < m Z ′ , while m l + l − and m τ + τ − are more evenly distributed in MSSM. This is because the underlying Drell-Yan processes are mostly mediated by the on-shell Z ′ in G2HDM, but by the off-shell SM γ and Z for MSSM, as indicated in Eqs. (9) and (10).
2. If the on-shell Z ′ is highly boosted and the mass splitting between Z ′ and the new leptons is large, one will have two collinear outgoing new leptons which result in two collinear SM leptons. In contrast, MSSM will not exhibit such a collinear behavior due to lack of Z ′ , and so one can distinguish G2HDM from MSSM via the event topology of dilepton plus missing transverse energy signals.
We display the corresponding cross section contours, similar to .01 pb as can be seen from the right column of Fig. 3.

V. FUTURE W ′ SEARCHES
In the event that Z ′ is discovered at HL-LHC via dijet or dilepton searches, one can ask whether it comes from an additional SU(2) gauge symmetry or simply from an extra U(1) ′ .
In this section, we discuss how to look for the electrically neutral W ′(p,m) whose existence will help to pin down the SU(2) H as a potential underlying symmetry in nature. In the rest of our analysis, we will switch back to Spectrum A and Spectrum B.  [18]. Both Z ′ and W ′ in LHT can only be produced in pairs due to T-parity and have the same signals just like W ′(p,m) in G2HDM. In other words, LHT is chosen as an illustrative example to underscore differences in the context of collider searches. Since W ′(p,m) in G2HDM is always heavier than Z ′ , it might not be easy to produce a pair of W ′(p,m) (or even Z ′ ) at the LHC, we will focus on the future 100 TeV proton-proton collider.
To identify SU(2) H unambiguously, the discovery of new heavy fermions Q H and L H as well as scalars like ∆ H and Φ H will also be necessary on top of W ′(p,m) and Z ′ .
We note that LRSM [17] has the right-handed charged W ± R . It can be singly produced and directly probed by dijet resonance [54,55] or same-sign dilepton plus two jets (l ± l ± jj) searches [56] depending on the right-handed neutrino mass, whereas W ′(p,m) in G2HDM must be pair produced. Due to quite different properties between W ′(p,m) and W ± R and the stringent bound on Z R : m Z R > 3.2 TeV [30], LRSM will not be considered here.
The LHT model discussed here is based on the coset manifold SU(5)/SO(5) which can be realized as a nonlinear sigma model [18]. Two different SU(2)×U(1) subgroups of SU(5) are gauged and are broken down to the SM electroweak gauge group SU(2) L × U(1) Y at a scale f T , which is higher than but not too far away from the electroweak scale so as to provide a possible solution to the fine-tuning problem. In the LHT model, all particles are divided into two classes based on the T-parity (denoted by P T hereafter), which corresponds to the symmetry under the exchange of the two SU(2)×U(1) subgroups. As a result, combinations of different fields of the two subgroups can be formed as having eigenvalue +1 or −1 of P T . The lightest T-odd particle is A H , a spin 1 particle, which is ensured to be stable and hence can be a DM candidate. Novel collider signatures like monojet and dijet plus missing transverse energy of A H in LHT was studied in [57]. All of the exotic particles have their masses proportional to f T since the masses are induced from the collective symmetry breaking at the scale f T . Furthermore, the exotic fermions couple to T-odd combinations of the gauge bosons of (SU(2) × U(1)) 2 . Three of the combinations comprise a non-abelian group SU(2) T which is broken at the scale f T . In the end, the exotic T-odd SU(2) T gauge bosons should couple to one T-even and one T-odd particles so as to conserve the T-parity.
We will use Z H and W ± H to denote the SU(2) T gauge bosons in the LHT, as opposed to Z ′ and W ′(p,m) in G2HDM. The quantum numbers of additional SU(2) L singlet fermions in the G2HDM and LHT are summarized in Table IV. Note that the superscript H is specifically  We will focus on leptonic decay channels for these gauge boson pairs because of the low QCD background as in the Z ′ resonance searches. The final states of two and four leptons plus the missing transverse energy, 2l + E T and 4l + E T respectively, will be investigated.
Take the W ′(p,m) pair in G2HDM as an example. The 2l + E T channel comes from the prompt decay of one of the two gauge bosons into one heavy charged lepton plus one SM lepton, while the other boson into one heavy neutrino plus one light neutrino. Each of the two resulting heavy fermions then decays into the DM particle H 0 2 plus the corresponding light SM fermion through the Yukawa couplings from (2). The neutrinos and DM particles in the final state will escape from the detector and manifest as E T . In Table V, we summarize the decay chains of the gauge boson pairs into 2l + E T and 4l + E T . In the last column labeled by Signal, the particles inside the curly brackets manifest as E T .

Model Production
Prompt Decay Final State Signal  Here we apply the same Spectrum A and Spectrum B for the gauge bosons, heavy exotic fermions and DM in G2HDM to the corresponding particles in LHT.
For G2HDM, the upper bound from the Z ′ resonance searches on the gauge coupling g H from Fig. 1 is used. In other words, the region above the line in each case is excluded.
That is the reason why the cross sections, which scale as g 4 H , increase when m W ′ (and also m Z ′ ) becomes larger, since the bound on g H becomes less stringent. By contrast, in LHT the gauge coupling is set equal to the SM electroweak coupling and thus the cross sections decrease as the gauge boson mass increases.
We should point out that in G2HDM the process pp → Z ′ → l H l H is actually the dominant contribution to the final state 2l + E T . If Z ′ is discovered in the resonance searches, as assumed here, one should be able to infer the precise values of g H and m Z ′ . Therefore, the dominant Z ′ contribution can be subtracted from the data so that one can study the contributions from pp → W ′p W ′m alone. Alternatively, as we shall see later one can also resort to the 4l + E T final state for W ′(p,m) searches. It is a relatively clean channel and is free from pp → Z ′ pollution. The corresponding cross section is only three times smaller than that of the 2l + E T channel. Finally, the Z ′ pair production pp → Z ′ Z ′ will also produce the 2l + E T and 4l + E T signals. The contributions, however, are at least two orders of magnitude smaller than those from pp → W ′p W ′m , and therefore will be neglected in our analysis.
In LHT, the cross section of pp → W + H W − H is about one order of magnitude larger than pp → Z H Z H at √ s = 100 TeV. That is because W + H W − H is produced dominantly by the s-channel γ and Z exchange, which is larger than the main contribution from the t-channel q H exchange to the Z H Z H production. Consequently, the cross section for 2l + E T , mostly from the W + H W − H channel, is almost one order of magnitude larger than that of 4l + E T , which arises only from Z H Z H channel. As seen from Fig. 4, however, the cross sections for both 2l + E T and 4l + E T are of the same order in G2HDM. On the other hand, the cross sections for both of these channels in LHT are roughly 1 ∼ 2 orders of magnitude larger than those in G2HDM, depending on the gauge boson mass. Thus, one can in general distinguish the two models just by measuring the total cross sections of the two channels.

D. The qualitative study: the kinematical distributions
In this section, we will further investigate the difference between G2HDM and LHT gauge boson decays in terms of three different normalized kinematical distributions. In principle, one should be able to distinguish the electrically charged W ± H from the neutral W ′(p,m) by the total charge of the corresponding decay products once they are produced singly. Both W ′(p,m) and W ± H , however, have to be pair-produced because of the SU(2) H and SU(2) T symmetry respectively which lead to the same total charge of the final states. As a consequence, the kinematical distributions of W ′(p,m) and W ± H decays are not only interesting but also important to study for further information, even though the production cross sections, as shown in previous Section, are in general much larger in LHT than in G2HDM.
Let X denotes W ′(p,m) , W ± H , or Z H , notwithstanding the same symbol has been used for the U(1) X gauge boson which has been assumed to be very heavy and decoupled. In four benchmark points m X = 0.5, 1.5, 3.0, and 4.0 TeV, we will show the normalized kinematical distributions of the spatial separation △R e + e − (Fig. 5) and invariant mass M e + e − (Fig. 6) of the electron pair in the 2l + E T channel, and the invariant mass M e + e − µ + µ − (Fig. 7) of four leptons in 4l + E T channel † † . The muon has the same distributions of △R µ + µ − and M µ + µ − as the electron, and will not be discussed separately. For a comparison, the benchmark point m X = 0.5 TeV is also included because its distribution shape is clearly distinguishable from the SM background. The corresponding coupling g H for m X = 0.5 TeV shall be appropriately small to avoid the current LHC limits as shown in Fig 1. The normalized kinematical distributions, nonetheless, do not depend on the values of g H that we shall keep in mind here and hereafter. In addition, the distributions do not change significantly between the two spectra and we simply choose Spectrum A. The leading order irreducible SM background for each kinematical distribution is also presented for comparison. Further discussions of these three distributions are as follows. † † The spatial separation between particles is defined as △R = (△η) 2 + (△φ) 2 , where △η and △φ are the difference in pseudo-rapidity and the azimuthal angle, respectively. On the other hand, the invariant mass squared between particles is defined as M 2 = (Σ i p i ) 2 , where p i is the four-momenta of particle i. Last but not least, the cross section of 4l + E T channel is about three times smaller than those of 2l + E T for W ′ (p,m) searches in G2HDM as can be seen in the left panel of Fig. 4.
Due to the facts that only W ′ (p,m) in G2HDM and Z H in LHT contribute to this channel and both of them are neutral, similar distributions of M e + e − µ + µ − between G2HDM and LHT exhibited in Fig. 7 are expected. However their distributions are clearly distinguishable from those of the SM which arise from on-shell Z decays and consequently peak toward low-energy regions. Thus final state of four leptons plus missing transverse energy can be used to detect physics beyond the SM. However to distinguish G2HDM from LHT is not easy using the 4l + E T channel unless M X is 0.5 TeV in which case a much smaller g H is anticipated.
From our studies of the two channels of two/four leptons plus missing transverse energy, one can conclude that in addition to the production cross sections, the kinematical distributions are also indispensable to discriminate the two models, G2HDM and LHT.

VI. SUMMARY AND OUTLOOK
One of the most interesting features in G2HDM [13], where the two The heavy fermion can decay into a SM fermion plus missing energy carried by the DM candidate H 0 2 , whose stability is protected by an emergent Z 2 symmetry. In this work, we studied collider signals of Z ′ and W ′(p,m) which can help us to pinpoint G2HDM. We derived constraints on Z ′ from the LHC 13 TeV data, followed by investigations of the future 14 TeV LH-LHC and 100 TeV proton-proton collider searches for Z ′ and W ′(p,m) .
The main difference between Z ′ and W ′(p,m) is that Z ′ can be singly produced and decay into SM fermions, while W ′ has to be pair-produced in light of the SU(2) H gauge symmetry.
It leads to stringent bounds on the SU(2) H coupling g H and the mass m Z ′ from the LHC direct searches based on the Drell-Yan type dilepton and dijet final states.
The updated LHC limit shown on (m Z ′ , g H ) plane in Fig. 1 is roughly a factor of two improvement in the low-mass region m Z ′ 2.25 TeV compared to the previous results [13] inferred from the LHC-8 dilepton data. In addition, for the whole region of m Z ′ of interest the dilepton constraints on g H now becomes more stringent than those deduced from LEP and the electroweak precision data. Even the dijet bounds which suffer from the QCD background are also stronger than the LEP bounds for m Z ′ 2.5 TeV.
Moreover, Z ′ can also decay into the new fermions which subsequently decay into SM fermions plus the missing transverse energy. We presented the contours of the rescaled cross sections σ/g 2 H on the plane of the masses of Z ′ and new heavy exotic fermions in Figs. (2) and (3) for Spectrum-A ′ and Spectrum-B ′ respectively. For Spectrum-A ′ , we considered final states of two leptons (es and µs) and two τ s with missing transverse energy which originate from decays of the exotic heavy leptons. For Spectrum-B ′ , two jets, two b-quarks and two t-quarks with missing transverse energy coming from the exotic heavy quarks were considered. Using the LHC constraints on ǫσ 95 obs from SUSY searches for the same final states, one can derive limits for the coupling g H from the contours of σ/g 2 H on the (m Z ′ , m l H ) or (m Z ′ , m τ H ) planes (Fig. (2)). While the constraints on g H obtained from exotic decays of Z ′ into heavy exotic leptons are comparable with those from the aforementioned dilepton and dijet searches, it appears that there is no severe bounds can be derived on the masses of heavy leptons from existing data of SUSY searches. On the other hand, for Spectrum-B ′ in which the pair production of the exotic quarks is kinematically allowed has substantial regions being excluded (Fig. (3)). The reason is that the dominant contributions to the exotic quark production arise from the QCD processes which are independent of the gauge coupling g H . Thus, stringent limits from LHC on SUSY squark searches can be directly applied to our case, requiring the new quarks to be heavier than 1 TeV or so.

Models
Production 2l If Z ′ can be discovered in the future, the neutral W ′(p,m) also need to be found in order to identify G2HDM as the underlying theory. Final states of two and four leptons with missing transverse energy from G2HDM and LHT had been studied in detail to underline different signatures between the neutral W ′(p,m) in G2HDM and the gauge bosons, Z H and W ± H , in LHT. The total cross sections for 2l + E T and 4l + E T were computed in both models (Fig. (4)). In the 2l + E T channel, the two final leptons come from the same W ′(p,m) in G2HDM, whereas in LHT they come from different W ± H . Therefore, a smaller spatial separation between the final leptons is expected for W ′(p,m) . Indeed as clearly seen in Fig. (5), the spatial separation of e + e − in LHT is completely overlapped with the SM one which has larger ∆R e + e − . Furthermore, by the same reason, the invariant mass of the lepton pair are cut off at the mass of W ′(p,m) as opposed to a flatter invariant mass distribution for LHT (Fig. (6)). In addition, the 4l + E T channel is also investigated. The invariant mass distributions of four charged leptons behave quite differently between the SM and G2HDM (and LHT), whereas LHT and G2HDM exhibit similar distributions (Fig. (7)).
We conclude our study by presenting the search strategies of distinguishing G2HDM from LHT in Table VI. For Z ′ which can be singly produced, the dilepton final states will be the best search channels. For pair-produced gauge bosons like W ′(p,m) in G2HDM, and W ± H and Z H in LHT, apart from the total cross sections, detailed kinematical distributions, such as (i) the spatial separation between the SM lepton pair and (ii) the invariant mass distributions of two and four leptons in the final states, can help us to disentangle these two models.
High luminosity upgrade for the LHC and building a future 100 TeV hadron collider are matters of utmost importance for fully exploring and distinguishing new electroweak scale models, all of which are contrived to address theoretical issues that cannot be answered within the SM. and dimensionless quartic couplings are necessarily real because every term in the mixed potential V mix (H, ∆ H , Φ H ) in (A7) is Hermitian.
The resulting coefficients of the quadratic terms for H 1 and H 2 after spontaneous symmetry breaking of SU(2) H induced by ∆ 3 = −v ∆ are where "· · · " refers to terms not containing v ∆ . As a consequence, even with a positive µ 2 H , the breaking of SU(2) H with v ∆ = 0 can trigger the breaking of SU(2) L to give rise H 1 = 0 if the sum of the second and third terms in (A8) can be sufficiently negative [33]. On the other hand, H 2 that does not develop a vev can play a role of the inert Higgs doublet.
where the three eigenvalues are in ascending order. Since we focus on the situation of v ∆ ∼ v Φ ≫ v, the lightest eigenstate with a mass m h 1 will be identified as the 125 GeV Higgs boson discovered at the LHC and the other two heavier Higgses h 2 and h 3 have the mass of m h 2 and m h 3 . The observed 125 GeV Higgs boson is a mixture of the three neutral components h, φ 2 and δ 3 .
The second mass matrix, comprised of three complex scalars G = {G p H , H 0 * 2 , ∆ p }, is quarks. The required Yukawa couplings y ′ f for new fermions in G2HDM are of order 10 −3 for Spectrum-A (A ′ ) and order 10 −2 for Spectrum-B (B ′ ) which are quite acceptable. On the other hand, the mass spectrum of the Higgses is the same for Spectrum-A and Spectrum-B and for Spectrum-A ′ and Spectrum-B ′ . Similarly cases for the mass spectrum of new gauge bosons. We should point out that the mass spectrum displayed here are composed of points in the parameter space that satisfy the two theoretical constraints -the scalar potential is bounded from below and all relevant 2 → 2 scattering amplitudes among the scalars are below the unitarity bound. In addition to the theoretical constraints, we also impose the experimental constraints from the 125 GeV Higgs, including its mass and signal strengths decaying into diphoton and τ + τ − . Detailed study of these constraints on the scalar sector of G2HDM is presented in a separate work [33]. One should be aware that we choose a parameter space slightly different from that presented in Ref. [33] where the new gauge sector is much heavier than the Higgs sector. In this analysis, we choose the new gauge bosons Z ′ and W ′(p,m) having masses between 1.5 TeV to 3 TeV such that they are accessible at the LHC. Nevertheless the parameter space in both works is chosen to satisfy the same set of aforementioned constraints.