Inert sextuplet scalar dark matter at the LHC and future colliders

We study a dark matter model constructed by extending the standard model with an inert $\mathrm{SU}(2)_\mathrm{L}$ sextuplet scalar of hypercharge 1/2. The sextuplet components are split by the quartic couplings between the sextuplet and the Higgs doublet after electroweak symmetry breaking, resulting in a dark sector with one triply charged, two doubly charged, two singly charged, and two neutral scalars. The lighter neutral scalar boson acts as a dark matter particle. We investigate the constraints on this model from the $\text{monojet} + /\!\!\!\!E_\mathrm{T}$ and $\text{soft-dilepton} + \text{jets} + /\!\!\!\!E_\mathrm{T}$ searches at the 13 TeV Large Hadron Collider, as well as from the current electroweak precision test. Furthermore, we estimate the projected sensitivities of a 100 TeV $pp$ collider and of a future $e^+e^-$ collider, and find that such future projects could probe TeV mass scales.

The standard model (SM) of particle physics is a self-consistent SU(3) C × SU(2) L × U(1) Y gauge theory describing the properties and interactions of three generations of fundamental fermions [1][2][3]. It has been well tested by a variety of experiments, and all the fundamental particles it predicts have been found. However, the SM cannot explain the existence of cold dark matter (DM) in the Universe, which is strongly suggested by astrophysical and cosmological experiments [4][5][6]. In order to account for particle dark matter, the SM must be extended.
Many popular extensions involve weakly interacting massive particles (WIMPs) as DM candidates, since thermal production of WIMPs in the early Universe is typically consistent with the observation of DM relic abundance. From the viewpoint of model building, WIMP models can be directly established by introducing extra colorless SU(2) L multiplets as the dark sector . The lightest mass eigenstate from the electrically neutral components of the multiplets could be a viable DM candidate, which can be either a fermion or scalar boson. In order to ensure the stability of the DM candidate, a Z 2 symmetry is commonly required.
In this work, we focus on the case of an inert sextuplet scalar with hypercharge Y = 1/2. A previous study [23] on this case investigated the phenomenological constraints from unitarity, electroweak oblique parameters, loop-induced Higgs decays into γγ and Zγ, DM relic abundance, and DM direct detection. A follow-up work [34] analyzed the impact of coannihilation and Sommerfeld enhancement effects on the relic abundance calculation, as well as the scale where a Landau pole appears. Here we extend these studies to the searches for the sextuplet scalar at the Large Hadron Collider (LHC) and future colliders.
After electroweak symmetry breaking, the mass eigenstates from the sextuplet scalar involve one triply charged scalar, two doubly charged scalars, two singly charged scalars, and two neutral scalars. The lighter neutral scalar boson acts as a DM candidate. At a high energy pp collider, the scalar bosons could be directly produced in pairs via electroweak interactions. Their decay chains would end at the DM candidate, which escapes detection and typically leaves a large missing transverse energy ( / E T ). Potential searching channels in pp collisions include the monojet + / E T and soft-dilepton + jets + / E T final states. We will investigate the constraints from current LHC searches. Moreover, the international high energy physics community is proposing future pp collider projects, such as the Super Proton-Proton Collider (SPPC) at √ s ∼ 70-100 TeV [61] and the pp Future Cir-cular Collider at √ s ∼ 100 TeV [62]. We will explore the prospect at a future 100 TeV pp collider based on simulation. Furthermore, electroweak interactions of the dark sector scalars also affect the electroweak oblique parameters at loop level, which can be examined in the electroweak precision test (EWPT). Future e + e − collider projects, such as the Circular Electron-Positron Collider (CEPC) [63], the e + e − Future Circular Collider [64], and the International Linear Collider [65], have plans to precisely measure the properties of the Z, W , and Higgs bosons and the top quark. These measurements would greatly improve the determination of the electroweak oblique parameters [66]. We will estimate the corresponding sensitivity.
The paper is organized as follows. In Sec. II, we introduce the inert sextuplet scalar model. In Sec III, we discuss the mass spectrum and decay processes. In Sec. IV, we investigate the current constraints from the monojet + / E T and soft-dilepton + jets + / E T searches at the LHC and the sensitivity at a 100 TeV pp collider. In Sec. V, we explore the current EWPT constraint and the future CEPC sensitivity. In Sec. VI, we give the conclusions.

II. INERT SEXTUPLET SCALAR MODEL
In the inert sextuplet scalar model, the SM is extended with a SU(2) L sextuplet scalar field S of hypercharge Y = 1/2 [23,34], which can be expressed as In the first line, the S components are labeled by their electric charges. Note that S must not be self-conjugated, implying S − = (S + ) † and S 2− = (S 2+ ) † . In the second line, we reexpress the components with the tensor notation, where the sextuplet is represented by a totally symmetric rank-5 SU(2) L tensor S ijklm with normalization factors ensuring The electrically neutral component S 0 can be divided into two real scalars φ 0 and a 0 , The lighter real scalar is a DM candidate. For ensuring its stability, we impose a Z 2 symmetry S → −S to make the sextuplet inert.
The Lagrangian involving the sextuplet scalar reads The covariant derivative of S is given by Here B µ is the U(1) Y gauge field and (W µ ) i j ≡ W a µ (σ a ) ij /2 is the SU(2) L gauge fields understood as a (1, 1) tensor for the SU(2) L group, where σ a denote the Pauli matrices. Thus, we have (W µ ) 1 The electroweak gauge couplings of the sextuplet components are given in Appendix A.
We write down the Z 2 -invariant scalar potential involving S as where H i is the SM Higgs doublet, and the rank-2 Levi-Civita symbol satisfies ij = ( As the quartic self-interaction terms of the sextuplet have no effects on the following discussions, we have not given their explicit expressions. If λ 3 is complex, say, λ 3 = |λ 3 |e iα , we can always absorb the phase α by redefining the sextuplet scalar field as e iα/2 S to make λ 3 real and positive. Without loss of generality, we use λ 3 ≥ 0 hereafter.
If the Z 2 symmetry S → −S is not respected, one may write down quartic terms like H † i S ijklm S † jknop S nopqr lq mr , which would lead to loop-induced decays of the DM candidate [23,67,68]. Therefore, such a Z 2 symmetry is not accidental, and we impose it by hand to stabilize the DM candidate.
After spontaneous breaking of the electroweak gauge symmetry, the λ 1 , λ 2 , and λ 3 terms in the unitary gauge become where λ ± ≡ λ 1 ± λ 2 and the VEV of the Higgs doublet is v = ( √ 2G F ) −1/2 . Thus, the Higgs VEV contributes to the masses of the sextuplet components. The mass terms read where the masses of the triply charged scalar S 3± and the neutral scalars φ 0 and a 0 are given by and the mass-squared matrices of the doubly and singly charged scalars are The gauge and mass eigenstates of the doubly and singly charged scalars are connected by rotations, with These rotations diagonalize the mass-squared matrices, resulting in masses given by Here we adopt a mass hierarchy convention of m 2 The rotation angles satisfy Note that λ + equally contributes to all the scalar masses, and mass splittings are only induced by λ − and λ 3 . Since we have adopted λ 3 ≥ 0, a 0 is not heavier than φ 0 . In order to prevent any charged scalar lighter than a 0 , we should require |λ − | ≤ 2λ 3 . In this case, a 0 is a viable DM candidate that cannot decay. The mass hierarchy of the dark sector scalars is Free parameters of the model are adopted to be The trilinear coupling between the Higgs boson h and the DM candidate a 0 is given by This coupling induces spin-independent (SI) a 0 scattering off nucleons in DM direct detection experiments. Making use of dimension-5 effective operators [69], we derive the a 0 -nucleon SI scattering cross section as where the form factors of nucleons f N u,d,s can be found in Ref. [70]. Note that the condition corresponds to flat directions among the scalar couplings, where the Higgs VEV gives no contribution to the a 0 mass and the ha 0 a 0 coupling vanishes. In this case, a 0 -nucleon scattering is absent at tree level, and direct detection experiments would hardly constrain the model.

III. MASS SPLITTINGS AND SCALAR DECAYS
If electroweak gauge symmetry is strictly respected, the components of the sextuplet scalar must have a common mass m S . Nonetheless, the VEV of the Higgs field breaks the degeneracy, leading nonzero mass splittings among the dark sector scalars. Figure 1  Mass splitting (GeV) m S for λ + = 1, λ − = 0, and λ 3 = 0.5. Note that m S universally contributes to all the scalar masses. As m S increases, the contributions to the masses from λ − and λ 3 become relatively small, and hence the mass splittings decrease. In Fig. 1(a), we also indicate the masses of the W ± and Z bosons to show whether W/Z is on-or off-shell in the scalar decays.
The largest mass splitting among the scalars is m φ 0 − m a 0 , whose contours in the m S -λ 3 plane are demonstrated in Fig. 1(b) for λ + = 1 and λ − = 0. We must ensure that the DM candidate a 0 has a physical mass, i.e., m 2 a > 0, corresponding to m 2 S > 3λ 3 v 2 /5 − λ + v 2 /4 − λ − v 2 /20. The hatched region in Fig. 1(b) is ruled out because it violates this condition. For m S 300 GeV, the mass spectrum is rather compressed.
Dark sector scalars heavier than a 0 are unstable. They can decay into lighter states through the mediation of the weak gauge bosons W ± and Z and the Higgs boson h. The mediation particles could be either on or off shell, depending on the mass splittings. Important decay channels include . The gauge and Higgs bosons subsequently convert to lighter SM particles. Typical 3-body decay diagrams are presented in Fig. 2. Because of the Z 2 symmetry, all decay chains finally end at the DM candidate a 0 . Typical mass splittings in this model are not large enough for an on-shell Higgs boson, and Higgs mediated decays into SM fermions are highly suppressed by the Yukawa couplings. Therefore, Higgs induced decays are commonly negligible.
For a scalar boson with 4-momentum p µ and mass m decaying into three particles with 4-momenta k µ 1 , k µ 2 , and k µ 3 and masses m 1 , m 2 , and m 3 , the partial decay width can be where |M| 2 is the invariant amplitude squared with summation over final state spins. The lower and upper limits of s 23 ≡ (k 2 + k 3 ) 2 are given by The lower and upper limits of s 12 ≡ (k 1 + k 2 ) 2 are s min 12 = (m 1 + m 2 ) 2 and s max 12 For decay processes of the dark sector scalar bosons, we derive the tree-level |M| 2 and calculate the partial widths and branching ratios utilizing the formulas above. For instance, the amplitude squared for the Z-mediated decay channel  where the 4-momenta are indicated between the parentheses, is obtained as Here c W ≡ cos θ W with θ W = tan −1 (g /g) denoting the weak mixing angle. Γ Z denotes the total width of Z, and g f V and g f A are the vector and axial-vector coupling coefficients of the SM fermion f to Z, respectively. On the other hand, the amplitude squared for the where Γ W is the total width of W and V ij is the Cabibbo-Kobayashi-Maskawa matrix.
In Figs. 3(a) and 3(b), we show the branching ratios of the decay channels of S + 1 and φ 0 , respectively. The parameters are fixed to be λ + = 1, λ − = 0, and λ 3 = 0.5, as the same as in Fig. 1(a).
The S + 1 boson only has W -mediated decay channels. For the parameters adopted in Fig. 3(a), the W boson is always off shell. In the S + 1 decay channels, the S + 1 → a 0 ud branching ratio B(S + 1 → a 0 ud) is the largest, while B(S + 1 → a 0 cs) is the second largest due to the suppression by the c and s quark masses. Because of the Cabibbo suppression, B(S + 1 → a 0 us) and B(S + 1 → a 0 cd) are negligible. B(S + 1 → a 0 e + ν e ) and B(S + 1 → a 0 µ + ν µ ) are basically identical, reflecting the lepton universality, while B(S + 1 → a 0 τ + ν τ ) is smaller, suppressed by the τ lepton mass. As m S increases, the mass splitting between S + 1 and a 0 decreases, leading to smaller phase spaces in the final states. Consequently, the S + 1 → a 0 cs and S + 1 → a 0 τ + ν τ channels are more suppressed for larger m S , resulting in larger B(S + 1 → a 0 ud) and B(S + 1 → a 0 + ν ), where = e, µ. Figure 3(b) demonstrates the Z-mediated φ 0 decay channels into a 0 as well as the Wmediated channels into S ± 1 . Because of the tiny mass splitting between φ 0 and S ± 2 , the φ 0 decay channels into S ± 2 are negligible and not shown in the plot. In the Z-mediated channels, B(φ 0 → a 0 νν) is as large as ∼ 20%. The φ 0 decay into a down-type (up-type) quark pair has a branching ratio ∼ 15% (∼ 11%), while the φ 0 decay into a charged lepton pair has a branching ratio ∼ 3%.
For a region around m S ∼ 165 GeV in Fig. 3(b), the Z-mediated channels commonly decrease; in contrast, the W -mediated channels commonly increase. This can be understood by comparing with the mass splittings shown in Fig. 1(a). For m S 160 GeV, we have m φ 0 − m a 0 m Z and the on-shell Z and W bosons dominate over the phase spaces. In this case, the φ 0 decays can be treated as 2-body decays φ 0 → a 0 Z and φ 0 → S ± 1 W ∓ , where the weak gauge bosons subsequently decay into SM fermions. For m S 160 GeV, however, the Z boson becomes off shell. Thus, the Z-mediated channels are suppressed by the phase spaces of 3-body final states. A similar suppression on the W -mediated channels happens later, at m S 185 GeV, where the behaviors of the two types of channels are gradually restored.

IV. SEARCHES AT pp COLLIDERS
High energy colliders are primary tools for finding new particles. Currently, the highest center-of-mass energy in pp collisions is √ s = 13 TeV, achieved by the LHC, which will be soon upgraded to √ s = 14 TeV. Moreover, a future pp collider with √ s ∼ 100 TeV would provide great opportunities for discovering the new scalar bosons in the inert sextuplet scalar model.
The dark sector scalars can be directly produced in pairs at pp colliders through electroweak gauge interactions. The inclusive production processes can be represented as . The corresponding Feynman diagrams at parton level are demonstrated in Fig. 4. The inclusive production cross section Typical Feynman diagrams for the scalar pair production at pp colliders, including as a function of m S is shown in Fig. 5 for λ + = λ − = λ 3 = 0, which implies that all dark sector scalars have a common mass m S . For m S ∼ 200 GeV, the cross section is ∼ 1 pb at √ s = 13 TeV, and increases to ∼ 30 pb at √ s = 100 TeV. For m S 700 GeV, the cross section increases by more than two orders of magnitude when √ s is promoted from 13 TeV to 100 TeV.
Since the only stable scalar is the DM candidate a 0 , all other scalars finally decay into a 0 , associated with visible particles. General detectors at the LHC and future pp colliders cannot probe a 0 , which would give rise to a missing transverse momentum / p T in the final state. The missing transverse energy is defined as / E T ≡ | / p T |, characterizing the energy amount of the particles that do not recorded by the detector. Therefore, a large / E T is an important imprint for a 0 . According to other information from the associating visible particles, we can utilize different channels to search for the dark sector scalars. In the following subsections, we separately discuss the monojet + / E T and soft-dilepton + jets + / E T search channels.
We will investigate the current LHC constraints on the inert sextuplet scalar model by reinterpreting the experimental analyses at √ s = 13 TeV and study the future sensitivity at a 100 TeV pp collider. For this purpose, we need to generate Monte Carlo simulation samples for the signals and relevant backgrounds. The inert sextuplet model is implemented by TeV and 100 TeV. The couplings are fixed as generate Feynman diagrams, calculate matrix elements, perform phase space integrations, and generate parton-level event samples. Furthermore, we use PYTHIA 8 [74] to carry out parton shower, hardronization, and particle decays. The MLM matching scheme [75] is adopted for merging matrix element and parton shower calculations. Finally, Delphes 3 [76] is used for a fast, simplified detector simulation with a parametrization of the ATLAS detector.
From the analysis in Sec. III as well as Fig. 1, we know that the mass splittings among the dark sector scalars are quite small for small scalar couplings and/or large m S . Therefore, in a large portion of the parameter space, the visible products from the scalar decays are quite soft. It is possible to utilize soft leptons (electrons and muons) for probing the signal, and we will discuss this possibility in the next subsection. In the present subsection, we choose to neglect the soft particles and consider at least one energetic jet originated from initial state radiation to recoil against the scalar pair for achieving a large / E T . Isolated leptons are further vetoed to give a clear signature. This leads to the monojet + / E T search channel, which has been widely used at the LHC for DM searches [41,55,[77][78][79][80][81].
The primary background in the monojet + / E T channel is Z(→ νν) + jets, where the neutrinos from Z decays cannot be detected and also lead to missing transverse energy. The subdominant background is W (→ ν) + jets, where the charged lepton could either fall outside of the detector acceptance or be combined into a jet [77]. Minor backgrounds include small fractions of Z/γ * (→ + − )+jets, multijet, tt+jets, single top, diboson (W W, W Z, ZZ) processes [82]. The multijet background can be efficiently reduced by requiring a sufficient large azimuthal angle ∆φ(j, / p T ) between / p T and each reconstructed jet j.

LHC constraint
Firstly, we utilize the monojet + / E T analysis from the ATLAS experiment with an integrated luminosity of 36.1 fb −1 at √ s = 13 TeV [83] to evaluate the monojet constraint on the model. In the ATLAS analysis, reconstructed jets are required to have p T > 30 GeV and |η| < 2.8, while reconstructed isolated electrons (muons) should have p T > 20 (10) GeV and |η| < 2.47 (2.5). Then the following selection cuts are applied for increasing the signal significance: (a) / E T > 250 GeV; (b) a leading jet with p T > 250 GeV and |η| < 2.4; (c) no more than four jets; (d) ∆φ(j, / p T ) > 0.4; (e) no isolated electron or muon. Moreover, various signal regions are defined according to further cuts on / E T , aiming at various models and different DM particle masses. In each signal region, the ATLAS collaboration derives the observed upper limit at 95% confidence level (C.L.) on the visible cross section σ vis , which is defined as the product of production cross section, acceptance, and efficiency. Here we choose several inclusive signal regions tabulated in Table I. Based on simulation, we estimate the visible cross section in these signal regions for the inert sextuplet model and derive experimental bounds.
The 95% C.L. exclusion region in the m S -λ 3 plane for λ + = λ − = 2λ 3 with the eight signal regions combined are shown in Fig. 6. Note that the parameter relation λ + = λ − = 2λ 3 satisfies the condition (26), and hence direct detection experiments give no constraint. Furthermore, this relation leads to degenerate mass spectra at tree level with Although electroweak loop corrections break the degeneracy and slightly lift up the masses of S ± 1 , S 2± 1 , and S 3± [8], the mass splittings between these scalar bosons and a 0 are too small to induce hard visible particles that can be triggered in monojet searches. Therefore, in this case these scalars also contribute to / E T , and one would expect a fairly well sensitivity in the monojet channel. Nevertheless, Fig. 6 shows that the 13 TeV monojet search only explore a region up to m S ∼ 85 GeV.  [83]. The blue and purple dashed lines denote the expected 95% C.L. exclusion limits at a 100 TeV pp collider with integrated luminosities 300 fb −1 and 3 ab −1 , respectively.

100 TeV pp collider sensitivity
Since the monojet + / E T search at the LHC does not explore the model well, we turn to study this channel at a 100 TeV pp collider and estimate the improvement of the sensitivity. We simulate signal and background event samples for pp collisions at √ s = 100 TeV. Only two dominant SM backgrounds, Z(→ νν) + jets and W (→ ν) + jets, are considered. Then we require reconstructed jets to have p T > 80 GeV and |η| < 2.8, and reconstructed electrons (muons) to have p T > 20 (10) GeV and |η| < 2.47 (2.5). Similar to the 13 TeV search, the following selection cuts are applied to the simulation samples.
• Cut 2.-No more than four reconstructed jets.
As illuminating examples, we consider four benchmark points (BMPs), whose detailed information is listed in Table II. All the four BMPs satisfy the condition (26), leading to m a 0 = m S , and there is no constraint from direct detection experiments. The mass spectra in these BMPs have various degrees of compression, which can be represented by the largest mass splitting m φ 0 − m a 0 . In order to estimate the sensitivity at a future pp collider, we define the signal significance as [84] where N S (N B ) is the number of signal (total background) events. β and γ represent the fractions of systematic uncertainties on N B and on N S , respectively. For the monojet channel at √ s = 100 TeV, we assume β = 1% and γ = 10%.
In Table III, we list the visible cross sections for the backgrounds and the four signal BMPs after each cut applied, as well as the signal significances of the BMPs. The veto on leptons in cut 4 reduces ∼ 39% of the W (→ ν) + jets background, since it often induces a hard lepton. The signal significances of the four BMPs remarkably increase after applying cut 5, reaching above 7. Overall, we find that these cuts efficiently suppress the backgrounds and increase S. Now we define four signal regions by separately requiring / E T > 1.6, 1.8, 2, 2.5 TeV. The expected exclusion limit at 95% C.L. combining these signal regions in the monojet channel with √ s = 100 TeV has been shown in Fig. 6 for λ + = λ − = 2λ 3 . We find that the 100 TeV monojet search could explore the parameter space up to m S ∼ 1.4 (1.65) TeV for an integrated luminosity of 300 fb −1 (3 ab −1 ).
As discussed above, the mass spectrum of the inert sextuplet scalar model is typically compressed. Consequently, leptons (electrons and muons) from off-shell Z and W bosons in the scalar decays are usually quite soft. Nevertheless, it is still possible to make use of such soft leptons for searching for dark sector scalars [85][86][87][88][89]. We will focus on a pair of same-flavor opposite-sign (SFOS) leptons (either e + e − or µ + µ − ) from off-shell Z decays, because they are rather unique for signal-background discrimination. The relevant scalar decay processes are S 2+ 2 → S 2+ 1 + Z * (→ + − ), S + 2 → S + 1 + Z * (→ + − ), and φ 0 → a 0 + Z * (→ + − ), whose Feynman diagrams are shown in Fig. 2(c) and branching ratios are exemplified in Table II. Some additional jets are allowed for keeping more signal events. In particular, one hard jet recoiling against the dark sector scalar pair is helpful for inducing a larger / E T and making the soft leptons more easy to be triggered. Thus, the search channel we would like to study is soft-dilepton + jets + / E T .
The important SM backgrounds in this channel are tt + jets, tW + jets, V V + jets (V = W ± , Z), and τ + τ − + jets. The top quark exclusively decays via the weak process t → bW . If the two W bosons from a tt pair decay leptonically, the final state would contain a pair of leptons associated with undetected neutrinos, mimicking the signal. The tW + jets background has a similar feature, but its production rate is lower. Nonetheless, a veto on b-tagged jets would be able to efficiently suppress both the tt + jets and tW + jets backgrounds. Moreover, the V V + jets and τ + τ − + jets backgrounds can contribute to the soft-dilepton + jets + / E T final state when the decays of the W/Z bosons or the taus produce a SFOS lepton pair.
Utilizing the information from the SFOS lepton pair and / p T , we can construct the m τ τ variable [88][89][90][91] for reducing the τ + τ − + jets background. When a tau pair recoils against an energetic jet, the taus are highly boosted and their decay products would be nearly parallel. In this case, for leptonic tau decays, the momentum of a daughter neutrino is basically collinear to the momentum of the corresponding daughter lepton. Thus, the 4-momentum of a neutrino ν i can be expressed as p µ ν i = ξ i p µ i , where p µ i is the 4-momentum of the related lepton i and the lepton mass has been neglected. Therefore, the total missing transverse momentum becomes / p T = ξ 1 p 1 T + ξ 2 p 2 T . By solving the two equations provided by this relation, we obtain the values of ξ 1 and ξ 2 for given / p T , p 1 T , and p 2 T . Since the 4-momentum of each tau is p µ τ i = p µ i + p µ ν i = (1 + ξ i )p µ i , the invariant mass squared of the τ + τ − pair can be expressed as m 2 τ τ = 2(1 + ξ 1 )(1 + ξ 2 ) p 1 · p 2 after neglecting the tau mass. The m τ τ variable is defined as m τ τ ≡ sign(m 2 τ τ ) |m 2 τ τ |. By this definition, m τ τ can be either positive or negative in practice. Nonetheless, m τ τ approximates the true invariant mass of a highly boosted τ + τ − pair with leptonically decays. We will see later that a veto on events with 0 < m τ τ < 200 GeV is helpful for suppressing both the τ + τ − + jets and V V + jets backgrounds at a 100 TeV pp collider.

LHC constraint
The soft-dilepton + jets + / E T channel has been used in the ATLAS search for electroweak production of supersymmetric particles with compressed mass spectra at √ s = 13 TeV with a dataset of 36.1 fb −1 [91]. We make use of the related search results to constrain the inert sextuplet model.
In the ATLAS search, reconstructed electrons (muons) are required to have p T > 4.5 (4) GeV and |η| < 2.47 (2.5), while reconstructed jets should have p T > 20 GeV and |η| < 4.5. The following selection conditions are further applied.
Then the ATLAS collaboration defines seven signal regions with different inclusive bins of m , and obtains the 95% C.L. observed upper limit on the visible cross section, as listed in Table IV.
Combining these seven signal regions, we derive the 95% C.L. exclusion region in the m Sλ 3 plane for λ + = λ − = 2λ 3 , as presented in Fig. 7. We find that the soft-dilepton+jets+ / E T channel at the current LHC is more sensitive than the monojet + / E T channel, excluding a region up to m S ∼ 210 GeV.

100 TeV pp collider sensitivity
Now we investigate the sensitivity of the soft-dilepton + jets + / E T channel at a 100 TeV pp collider. We require reconstructed leptons to have p T > 10 GeV and |η| < 2.5, and reconstructed jets to have p T > 40 GeV and |η| < 4.5. We apply the following selection cuts to the simulation samples.

Fraction of events
Soft dilepton, s = 100 TeV We find that the m 1 T and / E T /H lep T cuts used in the 13 TeV ATLAS analysis are not helpful for the inert sextuplet model, so these two cuts are abandoned for √ s = 100 TeV.
In order to justify these cut conditions, we demonstrate the distributions of backgrounds and signal BMPs in Fig. 8. These distributions are expressed as the fractions of events binned in kinematic variables. Moreover, the visible cross sections and the BMP signal significances are tabulated in Table V. For the soft-dilepton + jets + / E T channel at √ s = 100 TeV, we assume the fractions of the systematic uncertainties to be β = 1% and γ = 10%.
Firstly, Fig. 8(a) shows the ∆R distributions after applying cut 1. As we can see, the  tt + jets and tW + jets backgrounds tend to have much larger R than the BMPs and the other backgrounds. A part of the V V + jets background exhibits a similar behavior. This is because the two leptons come from decays of different particles in tt, tW , and W + W − and prefer to fly in opposite directions. For the BMPs and the V V + jets and τ + τ − + jets backgrounds, the SFOS lepton pair could come from an on-or off-shell Z boson, which is sufficiently boosted due to the requirements of / E T > 300 GeV and p j 1 T > 240 GeV in cut 1, and give small ∆R . Thus, the ∆R cut is rather useful for suppressing the tt + jets and tW + jets backgrounds, and also reduces the V V + jets background. From Table V, we find that the tt + jets, tW + jets, and V V + jets backgrounds lose roughly 44%, 47%, and 19% of events after we require 0.05 < ∆R < 2 in cut 2, respectively. Secondly, in Fig. 8(b), we present the m τ τ distributions after cut 2. As discussed above, m τ τ approximates the invariant mass of the tau pair in the τ + τ − +jets, resulting in a peak at m τ τ ∼ m Z . In addition, the tau pair from the W + W − → τ + τ − ν τντ process in the V V + jets background leads to a peak around ∼ 2m W . Therefore, the veto on 0 ≤ m τ τ ≤ 200 GeV in cut 3 significantly reduces the τ + τ − +jets and V V +jets backgrounds. As shown in Table V, only ∼ 19% and ∼ 58% of events in the τ + τ − + jets and V V + jets backgrounds remain after cut 3, respectively.
Thirdly, the distributions of the number of b-tagged jets after cut 3 are illustrated in Fig. 8(c). Since the jets induced by the b quarks from the tt+jets and tW +jets backgrounds have a high probability to be tagged as b-jets, most events from these two backgrounds have at least one b-tagged jet. As a result, the veto on b-tagged jets in cut 4 removes ∼ 63% (∼ 58%) of event in the tt + jets (tW + jets) background.
Finally, we demonstrate the m distributions in Fig. 8(d). As expected, the V V + jets background peaks around m ∼ m Z , manifesting the Z pole. Furthermore, the tt + jets and tW + jets backgrounds tend to have larger m than the BMPs. Consequently, the requirement of m ∈ [1, 3) ∪ (3.2, 70] GeV in cut 5 kills about 82%, 67%, and 63% of events in the V V + jets, tt + jets, and tW + jets backgrounds, respectively. On the other hand, since the SFOS lepton pair in the signal BMPs comes from the decay processes φ 0 → a 0 + − , S ± 2 → S ± 1 + − , and S 2± 2 → S 2± 1 + − , its invariant mass m reflects the mass splittings of the dark sector scalars. As shown in Table II, BMP1, BMP2, BMP3, and BMP4 have descending m φ 0 − m a 0 . Therefore, descending peaks exhibit accordingly in their m distributions.
In Fig. 9, we show the expected 95% C.L. exclusion limit in the soft-dilepton + jets + / E T channel with the six signal regions combined at √ s = 100 TeV, presented in the m S -λ 3 plane for λ + = λ − = 2λ 3 . With an integrated luminosity of 300 fb −1 (3 ab −1 ), the 100 TeV soft-dilepton search could probe a region up to m S ∼ 600 (840) GeV. Although the softdilepton channel is more powerful than the monojet channel at the 13 TeV LHC, it becomes less sensitive at a 100 TeV machine.

V. INDIRECT PROBE WITH ELECTROWEAK OBLIQUE PARAMETERS
Since the sextuplet components participate electroweak gauge interactions, they could contribute to several electroweak precision observables at one-loop level [23]. Most of the effects come from vacuum polarization diagrams of the electroweak gauge bosons and can be incorporated into the electroweak oblique parameters S, T , and U [93,94], whose values can be determined in the EWPT. In this section, we explore the current EWPT constraint on the inert sextuplet model, as well as the future sensitivity.
S, T , and U are linear combinations of the g µν coefficients Π P Q (p 2 ) of the gauge boson vacuum polarizations contributed by new physics, defined as where α is the fine structure constant, s W ≡ sin θ W , and Π P Q (0) ≡ ∂Π P Q (p 2 )/∂p 2 | p 2 =0 . Note that the SM predicts S = T = U = 0. The one-loop contributions to Π P Q (p 2 ) by the dark sector bosons in the inert sextuplet model are given in Appendix B.
A SU(2) L multiplet with nonzero hypercharge split by the Higgs VEV typically gives nonzero S, T , and U [95,96]. Nevertheless, if the interactions between the multiplet and the Higgs doublet respect a global SU(2) L × SU(2) R symmetry, a custodial SU(2) L+R symmetry [97] remains after electroweak symmetry breaking, resulting in vanishing T and U . In the inert sextuplet scalar model, the custodial symmetry corresponds to the condition To make this clear, we construct two SU(2) R doublet H I and S I (I = 1, 2), whose components are SU(2) L doublets and sextuplets, respectively, given by Thus, a generic potential respecting the global SU(2) L × SU(2) R symmetry can be written down as After electroweak symmetry breaking, such a potential would respect the custodial SU(2) L+R symmetry. As discussed above, the 4λ a H † i H i S † jklmn S jklmn term is not independent. Compared with the potential (6), we find that which lead to λ − = λ 1 − λ 2 = 2λ 3 . Therefore, λ − = 2λ 3 is a condition for respecting the custodial symmetry. On the other hand, if we define instead of Eq. (41), we would prove that λ − = −2λ 3 is another condition for the custodial symmetry.
In Fig. 10(a), we plot the electroweak oblique parameters S, T , and U as functions of λ − for m S = 500 GeV and λ + = λ 3 = 0.5. In this case, S increases as λ − increases, and T and U are not positive. At λ − = 0, S vanishes, while T and U reach their minimums. For λ − = ±1, the custodial symmetry condition λ − = ±2λ 3 is satisfied, leading to T = U = 0. In Fig. 10(b), the oblique parameters are presented as functions of λ 3 for m S = 500 GeV and λ + = λ − = 0.5. In this case, S hardly shows dependence on λ 3 , while T and U decrease as λ 3 increases. In both cases, U is always quite small, and this is typical in electroweak multiplet DM models [36,38]. Therefore, we neglect U in the following analysis.
Assuming U = 0, the Gfitter Group performs a global fit of current EWPT data and gives [92] S = 0.06 ± 0.09, T = 0.10 ± 0.07, ρ ST = 0.91,  [92]. The blue dashed lines denote the expected 95% C.L. exclusion limits of the future EWPT at the CEPC [63]. The red shade regions are excluded at 90% C.L. by the XENON1T direct detection experiment [98]. The red dashed lines indicate the expected 90% C.L. exclusion limits of the future LZ direct detection experiment [99].
where ρ ST is the correlation coefficient between S and T . We use this result to constrain the inert sextuplet model, as shown in Fig. 11. We find that the current EWPT excludes a region with m S 460 GeV in Fig. 11(a) for λ + = 1 and λ 3 = 0.5, and also excludes a region up to m S ∼ 900 GeV in Fig. 11(b) for λ + = λ − = 0.5. Moreover, in Fig. 7 for the relation λ + = λ − = 2λ 3 , a region with m S 140 GeV and λ 3 ≤ 0.5 is excluded by the current EWPT. Such a constraint is more stringent than the constraint from the 13 TeV monojet search, but looser than that from the 13 TeV soft-dilepton search.
Future e + e − collider projects could greatly improve the measurement of the electroweak oblique parameters, providing a powerful indirect probe to the sextuplet scalar. For instance, the CEPC EWPT could reach a precision of [63] σ S = 0.01, σ T = 0.01, ρ ST = 0.62, where σ S and σ T are the 1σ uncertainties of S and T , respectively. Figure 11 shows the expected 95% C.L. exclusion limits of the CEPC EWPT. We find that CEPC could reach up to m S ∼ 560 GeV in Fig. 11(a) and m S ∼ 1.08 TeV in Fig. 11(b). In Fig. 9 for the relation λ + = λ − = 2λ 3 , the CEPC EWPT could probe a region with m S 500 GeV and λ 3 ≤ 0.5.

VI. CONCLUSIONS
In this work, we have investigated a SM extension with an inert SU(2) L sextuplet scalar of hypercharge 1/2. After electroweak symmetry breaking, the quartic couplings between the sextuplet and the Higgs doublet split the sextuplet components. Thus, the mass eigenstates in the dark sector include one triply charged scalar S 3± , two doubly charged scalars S 2± 1,2 , two singly charged scalars S ± 1,2 , and two neutral real scalars φ 0 and a 0 . When the quartic couplings satisfy λ 3 ≥ 0 and |λ − | ≤ 2λ 3 , a 0 is the lightest dark sector scalar, acting as a viable DM candidate.
The mass spectrum in this model is typically compressed, leading to 3-body decays of the dark sector scalars mediated by the W , Z, and Higgs bosons. After pairs of these scalars are produced by electroweak processes at the LHC or a future 100 TeV pp collider, the resulting decay products are typically soft. This motivates us to consider the monojet + / E T and soft-dilepton + jets + / E T final states to search for the dark sector scalars at pp colliders. When the quartic couplings satisfy a particular relation (26), the DM-nucleon scattering is absent at tree level and DM direct detection experiments hardly probe the inert sextuplet model. In this case, collider searches provide complementary approaches to this model. Based on Monte Carlo simulation, we have derived the constraints from current LHC searches and have further evaluated the sensitivities at a 100 TeV pp collider. We have found that the 13 TeV monojet search excludes the parameter space up to m S ∼ 85 GeV, while the 13 TeV soft-dilepton search is more powerful, excluding the parameter space up to m S ∼ 210 GeV. At a 100 TeV pp collider, nonetheless, the monojet channel is much more sensitive than the soft-dilepton channel for probing the high mass region. This is because the mass spectrum becomes more compressed at higher mass scales, leading to softer decay products. With an integrated luminosity of 3 ab −1 at √ s = 100 TeV, the monojet (softdilepton) search could explore the parameter space up to m S ∼ 1.65 (0.84) TeV.
Since the sextuplet components contribute to the electroweak oblique parameters at oneloop level, EWPT provides an indirectly path to probe the model. We have found that the current EWPT excludes a paramter region with m S 900 GeV for λ + = λ − = 0.5, while the future CEPC EWPT could probe the parameter space up to m S ∼ 1.08 TeV.
Appendix B: Vacuum polarizations of electroweak gauge bosons In this appendix, we explicitly list the one-loop contributions to gauge boson vacuum polarizations Π P Q (p 2 ) by the dark sector scalar bosons in the inert sextuplet model. We express the results with the Passiano-Veltman scalar functions A 0 and B 00 [100], whose definitions are consistent with Ref. [101]. In our calculation, the numerical values of these functions are provided by LoopTools [102].
The photon vacuum polarization contributed by the dark sector scalars is where s 2+ ≡ sin θ 2+ and s + ≡ sin θ + . The Z boson vacuum polarization is given by , m 2 where c 2+ ≡ cos θ 2+ and c + ≡ cos θ + . The W boson vacuum polarization is