Almost Inert Higgs Bosons at the LHC

Non-minimal Higgs sectors are strongly constrained by the agreement of the measured couplings of the 125 GeV Higgs with Standard Model predictions. This agreement can be explained by an approximate $\mathbb{Z}_2$ symmetry under which the additional Higgs bosons are odd. This allows the additional Higgs bosons to be approximately inert, meaning that they have suppressed VEVs and suppressed mixing with the Standard Model Higgs. In this case, single production of the new Higgs bosons is suppressed, but electroweak pair production is unsuppressed. We study the phenomenology of a minimal 2 Higgs doublet model that realizes this scenario. In a wide range of parameters, the phenomenology of the model is essentially fixed by the masses of the exotic Higgs bosons, and can therefore be explored systematically. We study a number of different plausible signals in this model, and show that several LHC searches can constrain or discover additional Higgs bosons in this parameter space. We find that the reach is significantly extended at the high luminosity LHC.


Introduction
The discovery of the 125 GeV Higgs boson at the LHC [1,2] has been rapidly followed by an impressive program of measurement of Higgs couplings that tells us that the Higgs couplings are consistent with Standard Model predictions at the 10% level [3][4][5]. Further improving the Higgs coupling measurements is an important part of the ongoing physics program at the LHC and future colliders. An important complementary probe of the Higgs sector is direct searches for additional Higgs bosons. Additional Higgs multiplets are intrinsic to many extensions of the Standard Model that address the problem of naturalness, such as supersymmetry or composite Higgs models. In addition, from a purely phenomenological point of view, it is important to experimentally constrain non-minimal Higgs sectors that could play a role in electroweak symmetry breaking and the generation of elementary particle masses without reference to specific models of naturalness.
The consistency of the observed Higgs couplings with the Standard Model strongly constrain the possibilities for discovery of additional Higgs bosons. The simplest explanation for this consistency is that any additional Higgs multiplets have large positive electroweak-preserving mass terms. These models have a "decoupling limit" where the quadratic terms of the new Higgs fields get large, with other couplings held fixed [6,7]. In this limit, the physical masses of the new Higgs bosons becomes large, and their effects decouple at low energies. Probing additional Higgs bosons near the decoupling limit is therefore very difficult.
Another limit of multi-Higgs models that is often studied in the literature is the "alignment limit" where the lightest CP even physical Higgs boson h is closely aligned with the VEV in the multi-Higgs field space [7,8]. The decoupling limit implies the alignment limit, but alignment does not require the new Higgs bosons to be heavy. Alignment without decoupling is not guaranteed by any symmetry, and is therefore an accidental (or fine-tuned) property of the Higgs potential. The alignment limit has a distinctive phenomenology. The approximate alignment of the 125 GeV mass eigenstate h with the Higgs VEV guarantees that the couplings hV V (V = W, Z) are close to the Standard Model values. Since these are among the most precisely measured Higgs couplings, this partially explains the Standard-Model-like nature of the observed Higgs bosons. The alignment limit implies that couplings of the form HV V are suppressed, where H denotes a new Higgs boson. However, the couplings Hf f (f = fermion) are allowed to be unsuppressed, so one searches for signals involving the heavies fermions t, b, and τ [8][9][10].
In this paper we consider a simple symmetry explanation for the Standard-Model like couplings of the 125 GeV Higgs that allows additional Higgs bosons to be light. We assume that there are additional Higgs doublets that are odd under an approximate Z 2 symmetry, while all Standard Model fields (including the Standard Model Higgs doublet) are even under Z 2 . We first consider the limit where the Z 2 symmetry is exact, and then include small explicit breaking. First, note that the Yukawa couplings of the additional Higgs doublets to Standard Model fermions are forbidden by Z 2 symmetry. Next we consider the couplings of the Higgs bosons to vector bosons. We assume that the Z 2 odd Higgs fields have positive quadratic terms, so that they have vanishing VEV, and the Z 2 symmetry is not spontaneously broken. In this case the Z 2 -even and Z 2 -odd Higgs bosons do not mix, and the Standard Model Higgs doublet is entirely responsible for electroweak symmetry breaking. In this case, the vector couplings of the Z 2 even Higgs boson are the same as in the Standard Model, so the Z 2 symmetry gives a limit where the Higgs is naturally Standard Model-like. In this scenario, the additional Higgs bosons are called "inert" because they do not contribute to electroweak symmetry breaking [11]. In the inert limit, the lightest Z 2 odd particle is stable, and may be dark matter [12][13][14].
We consider the case where the Z 2 symmetry is approximate, so the new Higgs bosons are only approximately inert. We will assume that all Z 2 breaking terms are suppressed by a small dimensionless parameter . The parameter then suppresses single production of the new Higgs bosons, as well as their decays. Therefore, any deviation of the 125 GeV couplings to vectors or fermions from the Standard Model prediction is suppressed by , and the observed Higgs is naturally Standard Modellike.
The focus of this paper is on the collider signatures of these "almost inert" Higgs bosons. Standard searches for exotic Higgs particles at the LHC rely on single production of the Higgs particles, which is suppressed by in this scenario. For moderate values of (roughly < ∼ 0.1) these searches are completely ineffective due to low production cross-sections. However, couplings of the form V HH (V = W, Z, γ, H = exotic Higgs) are fixed by gauge invariance, and are unsuppressed in the inert limit. These are therefore the main production mode for the new Higgs particles. The decay of the new Higgs bosons to Standard Model particles is also suppressed by . This means that heavier Z 2 odd Higgs bosons will preferentially decay weakly to lighter Z 2 odd Higgs bosons, followed by a slower decay of the lightest Z 2 odd particle to Standard Model particles. This leads to cascade decays with multiple Standard Model particles in the final states. Although the last stage of the decays is suppressed by , it will still be prompt as long as > ∼ 10 −5 . Thus, for many orders of magnitude in the Z 2 breaking parameter (10 −5 < ∼ < ∼ 10 −1 ) the phenomenology is dominated by Type I 2HDM For H ± φ 0 and Aφ 0 the cross section depends on two independent masses; we choose m H ± = m A = 110 GeV and vary m φ 0 . prompt cascade decays, and this is the regime we will focus on in this paper.
In fact, the phenomenology of this model is almost completely determined by the masses of the Z 2 odd Higgs particles. The Z 2 symmetry allows electroweak symmetry violating mass splittings within the additional Higgs multiplets. (These arise from Z 2 invariant terms in the Higgs potential such as |H † 1 H 2 | 2 , where H 1,2 are the Z 2 even and odd Higgs doublets, respectively.) The leading production process for the new Higgs bosons is pair production from a virtual W , Z, or γ, namely 1 (1.1) The production rate for these processes is fixed by gauge invariance, and the rates at the LHC are shown in Fig. 1. The heavier new Higgs particles will generically have cascade decays to lighter members of the new Higgs multiplet by emitting a (possibly virtual) W or Z. These decays are not suppressed by , and therefore generically dominate over decays to Standard Model states. The lightest additional Higgs then has a "slow" decay only through Z 2 violating couplings. These can be thought of as arising from mixing with the Standard Model Higgs, and therefore go to the heaviest kinematically accessible Standard Model state. This gives rise to a rich set of manyparticle final states featuring the heaviest Standard Model particles: t, h, Z, W , b, and τ .
The decay cascades are generally dominated by a single decay mode at each stage of the decay, so the signal is determined completely by the masses of the new Higgs bosons. The lightest Z 2 odd Higgs boson decays to the heaviest kinematically available Standard Model particles. Weak production of Z 2 odd Higgs bosons can give H ± A 0 , H + H − , φ 0 H ± , or φ 0 A 0 . These then cascade decay down to the lightest Z 2 odd Higgs boson, generating a state with one or more vector bosons (W and/or Z) plus φ 0 φ 0 , H + H − or A 0 A 0 . The lightest Z 2 odd Higgs boson then decays to Standard Model particles. Because these decays occur via mixing with the Standard Model Higgs, these decays are to the heaviest kinematically accessible Standard Model final state. These decays are summarized in Fig. 2. Fig. 2 also shows the LEP bound on Z 2 odd Higgs bosons. LEP can directly produce φ 0 A and H + H − via Z * /γ * , which translates to a lower bound on the total mass of φ 0 A and H + H − . In addition, Fig. 2 gives a rough indication of the LHC reach for this model by showing the parameter space where the LHC production rate for a pair of Z 2 odd Higgs bosons becomes smaller than ∼ 1 fb. We also restrict ourselves to masses of φ 0 in the range 62.5 GeV < m φ 0 < 250 GeV (1.2) to avoid the processes φ 0 → hh and h → φ 0 φ 0 . Processes involving φ 0 → hh will be very challenging due to the low rate. The process h → φ 0 φ 0 dominates when it is kinematically accessible, but the phenomenology is different from the signals we consider here. We focus on the white region in Fig. 2, which illustrates the parameter space we are probing. The fact that this parameter space can be represented on a 2-dimensional plot means that the phenomenology of this scenario can be explored systematically.
We have investigated a large number of processes in this model that may be possible to probe at the LHC, listed in Tables 1, 2, and 3. The results of the investigation is summarized in the tables. There are a number of processes modes where a 5σ discovery is possible with 300 fb −1 for optimistic benchmark models. We will show below that there is significant additional parameter space that can be probed by the high luminosity LHC (3000 fb −1 ). The most effective searches are multi-lepton channels, due to relatively low Standard Model backgrounds. Illustrative event topologies leading to multi-lepton final states are shown in Fig. 3. Multilepton searches are standard parts of the LHC search program, so this establishes that this model will signal σ too small Table 2. Plausible channels assuming that m A = m H ± < m φ 0 and that φ 0 undergoes electroweak cascade decays. SSL means same-sign lepton pairs. OSSF means opposite-sign same-flavor lepton pair. Table 3. Plausible channels assuming that A, H ± , φ 0 undergo non-cascade decays. SSL means same-sign leptons. be probed by new LHC data. In addition, we identify one case where a novel search is sensitive, involving a lepton pair (opposite sign, same flavor) plus 3 b jets.
This paper is organized as follows. In §2 we give additional details of our benchmark model and its parameter space. In §3 we give details of several benchmark studies. §4 contains our conclusions, where we give projections of the search reach for both 300 fb −1 and 3000 fb −1 at the LHC.

The Model
We consider a model with 2 Higgs doublets H 1 , H 2 with an approximate Z 2 symmetry In the Z 2 symmetry limit, the Higgs potential is given by All couplings can be chosen real by rephasing H 1,2 , so this model naturally conserves CP . Note that the λ 3,4,5 terms can give unsuppressed mass splittings in the H 2 multiplet even in the Z 2 symmetry limit. We could even take the limit m 2 2 → 0, in which case all of the mass of the exotic Higgs bosons comes from electroweak symmetry breaking. In particular, the term |H 1 | 2 |H 2 | 2 contributes an electroweakpreserving mass for H 2 , which does not give rise to precision electroweak observables such as S and T . The fact that this mass comes from electroweak breaking is instead reflected in the fact that H 2 has large couplings to H 1 . Such large Higgs couplings are therefore the smoking gun signal of this kind of non-decoupling electroweak symmetry breaking. This particularly motivates the study of triple Higgs couplings in this class of models. We leave this study for future work.
We assume that m 2 2 > 0, so that in the Z 2 symmetry limit only H 1 gets a VEV. We then have where φ 0 , A, H ± are the physical fields that reside in H 2 . The custodial symmetry limit is λ 4 = λ 5 , which we assume from now on.
We also include O( ) terms that break Z 2 : Not all of the couplings in Eqs. (2.2) and (2.4) are important for phenomenology. This is because H 2 = O( ), and we are not interested in terms with more than 2 Higgs fields. The effects of λ 2 and ∆λ are therefore suppressed by , and we can neglect them to get an overview of the phenomenology. (We can think of H 2 as "small.") Since we also set λ 4 = λ 5 , we effectively have 7 parameters instead of 10: The first two parameters are of course fixed by experiment to be m h = 125 GeV and v = 246 GeV, leaving 5 free parameters. However, we will show that for small the phenomenology is essentially determined by the mass spectrum of the new Higgs bosons.
Production of Z 2 odd Higgs bosons comes from the couplings such as g ZAφ 0 , g ZH + H − , and g W + H − φ 0 , which are fixed by gauge invariance. Decays of heavier Z 2 odd Higgs bosons to lighter Z 2 odd Higgs bosons are controlled by the same couplings. The only additional couplings that we need are the ones that determine the decay of the Z 2 odd Higgs bosons to the Z 2 even Higgs bosons and Standard Model vector bosons. For these we must consider the minimization of the Higgs potential.
We define the physical fields h, φ 0 , A, H ± in terms of the fields The physical pseudoscalar field is then given by while the physical scalars are Using standard results from 2 Higgs doublet models, together with g h 1 V V ∝ v 1 and Eq. (2.8), we then obtain the interaction vertices that control the decays of the lightest Here the 4-momenta are all defined to flow into the vertex. We now discuss couplings of the Z 2 odd Higgs bosons to fermions, which are relevant for the decay of the lightest Z 2 odd Higgs boson. We define the fermions to be even under Z 2 , so in the Z 2 symmetry limit, only one Yukawa couplings to H 1 are allowed. This is a "type I" 2-Higgs doublet model, which naturally avoids non-Standard Model flavor violation. When we include Z 2 breaking, we must allow O( ) Yukawa couplings to H 2 , so this models is no longer type I for = 0. We then have to worry about re-introducing unacceptably large flavor violation at O( ). It may be interesting to consider the possibility that sufficiently suppresses non-Standard Model flavor violation. Our focus is on direct searches for new Higgs bosons, so we will avoid flavor problems by making the phenomenological assumption that all flavor breaking is contained in a single set of Yukawa coupling matrices y u , y d and y e . This is "minimal flavor violation." Its validity depends on the UV completion of the theory having a single source of flavor breaking, at least to a very good approximation. With this assumption, the couplings of the Higgs fields to fermions is given by We will also make the phenomenological assumption that u d e . (2.14) Then we have for any fermion f We see that the decays of φ 0 , A and H ± to fermions is controlled by the small parameter It is natural to assume that f ∼ V . Note that both V and f involve h , which depends on Z 2 breaking in the Higgs potential. Therefore it is not natural to have V f . If we have f V , then fermion loops will induce Z 2 breaking in the Higgs potential. For the top quark loop, we expect where Λ is a UV cutoff. Even for Λ ∼ TeV this is not suppressed.
Although we will assume f ∼ V in our study, the relative size of these suppressions is important for phenomenology because it determines the masses at which different decays become dominant. For example, if φ 0 is the lightest Z 2 odd Higgs boson, it can decay either to W W or bb. The decay to bb becomes dominant for m φ 0 < ∼ 2m W , but the precise mass for which this occurs is sensitive to the ratio (2.18) Fig. 4 shows branching ratios of the main decay modes of φ 0 , A and H ± to the SM particles for r = 1/5 and the dashed lines assume that r = 5. The phenomenology therefore depends on this parameter in addition to the spectrum of Z 2 odd Higgs bosons. This parameter affects only the reach of a given search, so searches can be optimized only on the basis of the spectrum of masses of the exotic particles.
For all the benchmark models considered in our paper, we found parameters in the 2-Higgs doublet model parameter space that give an experimentally acceptable contribution to the S parameter. This is easily accomplished despite the fact that the additional Higgs bosons are light because they are approximately inert. In addition, these models easily satisfy all perturbativity constraints on the potential because the additional Higgs bosons are all light.

Benchmark Studies
In this section we study several benchmark models with multi-lepton signals. Simulated events for both signal and Standard Model backgrounds were generated by MadGraph5 [18], with showering and hadronization simulated by Pythia8 [19], and the detector response simulated by Delphes3 [20]. The leading order cross-sections of the signal and Standard Model backgrounds for each channel are calculated by MadGraph5. Several of the Standard Model backgrounds, such as tt and W/Z+jets have large NLO contributions, therefore we scale the LO cross sections of these processes with their corresponding K-factors [21]. Since we focus on the final states that contain leptons and b jets, common selection requirements are applied to reconstructed jets, muons and electrons, before further selection requirements, optimized for each final state, are applied. Leptons are required to have a transverse momentum p T > 10 GeV and pseudorapidity |η| < 2.5. We further require isolated leptons, as determined from the isolation ratio R iso = p T j /p T where p T j is the clustered transverse energy, contained in a cone of radius ∆R around the lepton, and p T is the lepton transverse energy. The lepton isolation requirement used in this analysis is ∆R < 0.2 with R iso < 0.09. Similar isolation criteria have been used by ALTAS for their multilepton searches in LHC Run II [22].
Jets are required to satisfy p T > 20 GeV and |η| < 5. The b-tagging efficiency is taken to be the same as the default setting in Delphes3. The remaining event selection is optimized for each individual channel, as described below.

3 leptons off Z Peak
In the case when φ 0 , H ± and A are all relatively heavy, they dominantly decay to final states that contain W or Z. In particular, H ± (A) can decay to W (Z)φ 0 or W (Z)h depending on the mass splitting between φ 0 and H ± (A). In this scenario, pair-produced non-Standard Model H's can decay to five to six on-or off-shell vector bosons (Figure 3a, 3b), therefore easily producing multiple leptons in the final state.
Asking for 3 light leptons has the advantage of a relatively low Standard Model background at LHC. Furthermore, given that pair produced φ 0 H ± , φ 0 A, AH ± and H + H − may all contain 3 leptons in their final states, this channel also benefits from high signal multiplicities. Its drawback is that signal decays cannot be reconstructed, hence the signal kinematic features are not prominent enough to discriminate them against SM backgrounds. As a result, this channel basically becomes a lepton counting channel, which can be potentially covered by the 3-lepton bin of general multi-lepton searches from ATLAS and CMS. Figure 5 shows the main result of this search, where we draw the 5σ contours reached at LHC run II and high-luminosity (HL) LHC. As we shall see, the overall 5σ reaches are not affected by varying s as long as f V . The reason is that for any values of the s considered, the final states of the exotic H's decays always include combinations of the SM vector bosons and 125 GeV Higgs. Table 4 lists three benchmarks that are representative of each type of decays based on the assumptions on the mass hierarchy of φ 0 , A and H ± . Benchmark 1(B1) gives an example of the scenario in which m φ 0 < m H ± ,A and the mass splitting between φ 0 and H ± (A) is sufficient to allow H ± (A) to decay to W (Z)φ 0 . This corresponds to the Table 4. Details of the benchmarks for the 3 leptons off Z Peak search.
region below the diagonal line in Figure 5. Moving towards the diagonal, the mass splitting shrinks and the dominating decay modes of H ± (A) is to go through W (Z)h as long as V > f (B2). As we cross the diagonal, where The main Standard Model backgrounds include dibosons, ttV , and tt or Z plus jets with one fake/non-prompt (FNP) lepton. To estimate the FNP leptons, we simulate Z plus jets and tt, both of which are then decayed to include at least two leptons. We then select events that contain at least two reconstructed leptons and one jet, assuming a flat jet-faking-lepton rate. We match their contributions to the 3 bin in Figure 2(d) of the 36 fb −1 ATLAS multi-lepton search [24] and extract the jet-faking-lepton rate ∼ 8 × 10 −4 .
For the preselections, we require b-veto, at least 3 leptons with p T of the leading (sub-leading) lepton > 20 (15) GeV. If a pair of OSSF leptons are found, we require that their invariant mass / ∈ (m Z − 15, m Z + 15) GeV. Since the signals are relatively massive and typically are five or six vector bosons with more than half of them undergoing leptonic decays, we also require that missing energy / E T > 40 GeV and H T > 300 GeV, where H T is the scalar sum of the lepton and jet p T 's.
Due to the limited signal numbers and a lack of prominent kinematic features of them, we are only able to place a final cut on the number of jets, N j . Table  5 gives the yields of the signals and the SM backgrounds assuming an integrated luminosity of 300fb −1 . Only B1 signal can reache 5σ significance. For B2 (B3) signals, approximately 2000fb −1 (2500fb −1 ) is required to achieve a 5σ significance. Compared to the previous case, B2 and B3 perform much worse, mainly because h → V V is not the dominant decay mode for 125 GeV Higgs.
From the benchmark studies, it can be seen that in the case of a small r(≡ f / V ) and large Higgs masses, φ 0 , A and H ± dominantly decay to V + X. Regardless of what assumptions are made about their mass hierarchy, the pair produced exotic Hs can always contribute to the signal 3 off Z. We also investigate whether our results will be affected by varying the absolute values of s. In The signal region is defined by b-veto, / E T > 40 GeV, HT> 300 GeV and N j > 2. The 3000 fb −1 limit is further divided into three subregions where more than half of the signals come from each of the 'colored' decays.
decays of φ 0 , H ± and A can be different under the variation of V , they all end up contributing to the signals that we are looking for. As a result, the 5σ limit contour does not depend much on the absolute values of V or f . As long as V is much larger than f , the three types of decays compliment each other. σ(fb) Table 5. Signal and the background yields for the channel 3 off Z, assuming an integrated luminosity of 300 fb −1 . To estimate the number of events with FNP leptons, a flat jet-faking-lepton rate of 8 × 10 −4 is used. The preselections are 3 off Z, b-veto, / E T > 40 GeV, HT> 300 GeV and the final selection is N j > 2.

OSSF leptons with 3 b jets
The 3 off Z search above targets the parameter space with relatively massive Z 2 odd Higgs particles. In this section, we look at a relatively light φ 0 ( 120 GeV), where φ 0 → bb becomes the dominant decay mode.
If (m H ± =)m A > m φ 0 , A predominantly decays to φ 0 Z ( * ) . One interesting channel to consider is depicted in Figure 3c, where pp → φ 0 A → φ 0 (φ 0 Z ( * ) ) → (bb)(bb + − ) gives a final state that consists of a pair of opposite-sign same-flavor (OSSF) leptons and four bs. Therefore, we ask for a pair of OSSF leptons with the leading (subleading) lepton p T > 20(15) GeV, and at least 4 jets with 3 b-tagged jets. Since there is no invisible particles for the signal process, we also require / E T < 50 GeV as part of the preselections.
The dominating SM backgrounds are Z+jets, di-leptonic tt and single top production. Other SM backgrounds include di-bosons, V h and fake/non-prompt leptons, but they are negligible compared to the first three SM processes [23].
Depending on whether the mass difference between A and φ 0 is greater than 91 GeV or not, this channel is further divided into the on-and off-shell Z signal regions. Below we give detailed benchmark studies focusing on each region. For both choice of benchmarks, we further assume that r ≡ f / V = 5, f = 0.1. Under these assumptions, BR φ 0 →bb is approximately 80% and BR A→Zφ 0 almost 100%. After applying the preselections discussed above, we try to reconstruct the entire decay chain for the signal. Since both φ 0 s decay to bb, we assume that the jet with the highest transverse momentum out of the non-b-tagged jets to be the fourth b. To reconstruct the φ 0 s, we choose the combination of the jets that minimizes (∆φ j 1 ,j 2 ) 2 + (∆φ j 3 ,j 4 ) 2 . Since A decays via φ 0 and Z ( * ) , we then reconstruct A using the combination of the two leptons and the reconstructed φ 0 that has a smaller value in |∆φ| . Figure 6 shows the reconstructed A and φ 0 mass distributions for signal and backgrounds after the preselections. As can be seen, both show prominent resonances for the signal, hence can be used to effectively suppress the backgrounds. The Table 6 gives the yields of the signal and the dominating backgrounds assuming an integrated luminosity of 300 fb −1 . To achieve a significance of 5σ, we need approximately 700 fb −1 . This benchmark produces an on-shell Z in its decay, therefore we apply the same preselections as before except for requiring on Z instead of off Z. We repeat the analysis as above. The final selections are / E T / √ HT < 2 GeV 1/2 , |m bb − m φ 0 | < 20 GeV and m bb ¯ − m A < 10 GeV. Table 6 gives the signal and background yields assuming an integrated luminosity of 300 fb −1 . To achieve a significance of 5σ, we need roughly 2000 fb −1 . The onZ case performs much worse compared to the offZ case from the two benchmark studies.  Table 6. Signal and background yields for OSSF leptons plus 3 bs assuming an integrated luminosity of 300 fb −1 . The signal benchmarks both have f = 5 V = 0.1. The preselections for Benchmark 1 (2) are OSSF pair off (on) Z, N j > 3 with at least 3 b tagged and / E T < 50 GeV. The final selections for B1 and B2 are as described in the text above.

2 Same-Sign leptons
The search channel above targets a light φ 0 . In this subsection, we consider a light H ± (A). If m φ 0 > m H ± (= m A ), φ 0 → H ± W ∓ ( * ) or AZ * become the dominant decay. If H ± is lighter than 130 GeV, it decays to τ ν predominantly. As depicted in Fig. 3d, where pp → φ 0 H ± → (H ± W ∓ )H ± with H ± → τ ν, if W further decays leptonically, we can easily obtain a final state of ± ± or τ ± h ± , where represents e or µ and τ h a τ -tagged jet.
For this search, we only consider the final states µ ± µ ± or µ ± τ ± h . τ ± h τ ± h is not included because it suffers from a huge multi-jet background without light leptons. Electrons are not considered here because the charge misidentification is non-negligible for electrons. The benchmark we choose to work with is (m φ 0 , m H ± ) = (160, 110) GeV with r = 1/5, V = 0.1, where BR φ 0 →H ± W ∓ is 80% approximately and BR H ± →τ ν 65% approximately.
The main irreducible backgrounds are dibosons, ttV , V V V . The SM backgrounds with one fake/non-prompt (FNP) lepton or one fake τ h come from W or Z plus jets and tt. The fake rate is estimated to be approximately 10 −4 .
For preselections, we ask for two same-sign muons or one muon plus one same-sign τ -tagged jet. Events that have any bs are vetoed. We further require that / E T > 85 GeV, because the signal has multiple invisible particles in its final state.   To combat the W Z and W +jets backgrounds, we look at the transverse mass of the W : Since there are two leptons, we reconstruct m W T for both of them and take the smaller one to be m W T . Based on the kinematic distributions plotted in Figure 7, the final selections comprise 7 > N j > 2, ∆φ > 2.1 and |m W T − m W | > 5 GeV. Table 7 gives the yields of the signal and background processes assuming an integrated luminosity of 300 fb −1 . To get 5σ, an integrated luminosity of 600fb −1 is required.

Conclusions
In this paper we considered the phenomenology of a 2-Higgs doublet model where the additional Higgs bosons are almost inert. This means that there is an approximate Z 2 symmetry that ensures that there is a Standard Model-like Higgs boson mass eigenstate whose VEV is dominantly responsible for the masses of Standard Model vector bosons and fermions. This fully explains the agreement of the couplings of the observed 125 GeV Higgs boson, while allowing the additional Higgs bosons to be light and therefore kinematically accessible at the LHC. The phenomenology of this kind of model is very distinctive. The Z 2 odd Higgs bosons are pair produced  Table 7. Signal and background yields for the same-sign leptons assuming an integrated luminosity L = 300 fb −1 . To estimate the FNP leptons, we use a flat fake rate to be ∼ 10 −4 . The signal benchmark is that m H ± = 110 GeV, m φ 0 = 160 GeV and r = 1/5, V = 0.1. The preselections are SS µµ or µτ h , b-veto and / E T > 85 GeV. The final selections are 7 > N j > 2, ∆φ > 2.1 and |m W T − m W | > 5 GeV.
by electroweak interactions, and undergo cascade decays with the heaviest Standard Model states at the end of the decay chain.
In this paper we initiated the exploration of the phenomenology of this class of models. We focused on LHC searches, and showed that these are sensitive despite the low production cross sections. The most effective searches are multi-lepton searches, but custom searches involving leptons and b jets are also effective. Figure 8 summarizes our results. We show the 5σ reach for each search for an LHC integrated luminosity of 3000 fb −1 (dashed) and 300 fb −1 (solid). We also compare the bounds with those from future e + e − colliders, which will be both clean in the background and efficient in producing the types of signals we study here. We conclude that the high luminosity LHC can explore a significant region of the parameter space of these well-motivated models.