Type-II seesaw scalar triplet model at a 100 TeV pp collider: discovery and higgs portal coupling determination

We investigate the collider phenomenology of the scalar triplet particles in the Type-II seesaw model at a 100 TeV pp collider. Depending on triplet vacuum expectation value vΔ, the dominant discovery channels could be H++H−− and H±±H∓. We find the H±±H∓ → W±W±hW∓/ℓ±ℓ±hW∓ channels are promising for both model discovery at relatively large vΔ and determination of the Higgs portal couplings λ4 and λ5. We also find that these two channels are complementary to indirect determination of λ4 from future precise measurements on h → γγ decay rate. Together with pair production of the doubly-charged Higgs subsequently decaying into same-sign di-leptons, the H±±H∓ channels have the potential to cover a significant portion of the parameter space of the Type-II seesaw complex scalar triplet model.


Introduction
Explaining the origin of neutrino masses is a key open problem in particle physics. The significant difference in magnitudes between the masses of the charged and neutral leptons suggests that the dynamics responsible for the observed light neutrino masses, generically denoted here as m ν , are different than those of the Standard Model (SM) Higgs mechanism. Among the most widely considered is the seesaw mechanism. Its theoretical attractiveness rests in part on the idea that the suppression of m ν results from a ratio of physical scales rather than the appearance of tiny dimensionless Yukawa couplings in the Lagrangian. Several variants of the seesaw mechanism have been studied over the years, with perhaps the types I, II, and III models [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] the most thoroughly considered.
It remains to be seen which, if any of these scenarios, is realized in nature. In the conventional type-I model [1][2][3][4][5], the scale of the heavy, right-handed (RH) Majorana neutrinos, M N , lies well above the energies directly accessible in the laboratory, making a direct probe of this scenario infeasible. Theorists have considered lower scale variants with M N at the TeV scale or below, a possibility that allows for more direct experimental tests, including the observation of the RH neutrinos in high energy collider searches or beam dump experiments. In this case, the scale of the relevant Yukawa couplings need not be too different from those of the charged leptons.
In this study, we consider the type-II scenario [6,[8][9][10][11]18], wherein the scale of m ν is governed by the product of Yukawa couplings h ν and the vacuum expectation value (vev) v ∆ of the neutral component of a complex triplet ∆ that transforms as (1,3,2) under the SM gauge groups. Constraints from electroweak precision tests require that v ∆ be no larger than a few GeV, though it could be considerably smaller. Consequently, the Yukawa couplings h ν may be as large as O (1). As in the case of low scale type I models, the mass scale of the ∆ may lie at the TeV scale or below without introducing new naturalness issues beyond those already present in the SM Higgs sector. It is, then, interesting to ask under what conditions one may discover the new degrees of freedom essential to the type II scenario and to what extent its interactions determined.
In this study, we focus on these questions, paying particular attention to the ∆ interactions in the scalar sector. With the discovery of the SM-like Higgs boson [19,20], it is JHEP01(2019)101 timely to consider the scalar sector potential in more detail. In general, the presence of additional scalar degrees of freedom that interact with the Higgs doublet Φ may enhance stability of the potential, as has been noted in the case of the ∆ in refs. [21][22][23][24]. In addition, ∆-Φ interactions may allow for a strong first order electroweak phase transition (SFOEWPT), thereby providing the needed conditions for generation of the cosmic baryon asymmetry through electroweak baryogenesis. 1 In both cases, knowledge of the Higgs portal couplings λ 4 and λ 5 (defined below) is essential. This study represents our first effort to provide a roadmap for discovery of the ∆ and determination of its scalar sector couplings, building on the results of earlier studies that focus on the collider phenomenology of the ∆ at LEP and the LHC 2 as well as its impact contributions on the SM Higgs di-photon decay rate [22,.
Searches for the complex triplet scalars -including doubly charged H ±± , singly charged H ± , and neutral Higgs particles H and A -have been carried out at the LHC. A smoking gun for the CTHM has conventionally been the presence of the H ±± decaying into a same-sign di-lepton final state and has been intensively investigated by the ATLAS and CMS collaboration [122][123][124][125][126][127][128][129][130]. For other channels related to CTHM discovery, there are also many studies have been done at the LHC, see appendix A for a detailed summary.
In what follows, we explore the potential for both discovery of the ∆ and determination of its scalar sector couplings at a prospective future 100 TeV proton-proton collider, such as the Super Proton Proton Collider (SppC) under consideration in China and the CERN Future Circular Collider (FCC-hh). Given the higher center of mass energy and prospective integrated luminosity, a 100 TeV pp collider will provide coverage for a considerably larger portion of model parameter space than is feasible with the Large Hadron Collider (LHC). In this context, there exist two distinct mass spectra for the ∆ (governed by the model parameters), as discussed in detail in section 3.1. By working in the "normal mass hierarchy", where m h ≤ m H/A ≈ m ∆ ≤ m H ± ≤ m H ±± with m ∆ the mass scale of the model, we find that: • The future 100 TeV pp collider with an integrated luminosity of 30 ab −1 can discover the triplet model up to m ∆ 4.5 TeV for v ∆ ≤ 10 −4 GeV and m ∆ 1 TeV for v ∆ 10 −4 GeV. Our result is shown in figure 7.
• Upon discovery, the Higgs portal parameter λ 5 can be determined from the mass spectrum of H ±± and H ± for m ∆ 1 TeV, while λ 4 is determined by the branching ratio (BR) of H ± → hW ± . The h → γγ decay rate also provides a complementary probes of the related parameter space, as we discuss below in relation to figure 11.
In our analysis leading to these conclusions, we first study the same-sign di-lepton decay channel for pp → H ++ H −− , whose production cross section at √ s = 100 TeV is the largest among all triplet scalar channels. We find that this channel is only suitable for the triplet model discovery at small v ∆ , where the corresponding Yukawa couplings h ν that JHEP01(2019)101 govern the H ±± decay rate can be relatively large and still consistent with the scale of m ν . For relatively large v ∆ , we find that there exist other promising discovery channels, particularly pp → H ±± H ∓ with H ±± → W ± W ± / ± ± and H ∓ → hW ∓ . Considering these channels at both small and large v ∆ will allow for discovery over the entire range of v ∆ parameter space for triplet mass up to ∼ 4.5 TeV (∼ 1 TeV) as can be seen from figure 7.
Assuming discovery, the next question we ask is: how does one determine the Higgs portal couplings? We find that measurement of the rate for pp → H ±± H ∓ with H ±± H ∓ → W ± W ± hW ∓ / ± ± hW ∓ W ∓ decaying leptonically will be advantageous. These two channels probe a signifiant portion of the relevant entire parameter space as can be seen from figure 11. The presence of the charged triplet scalars with masses and couplings in the same range could also lead to an observable deviation of the h → γγ signal strength compared to Standard Model expectations. For triplet scalar masses below roughly one TeV, the prospective future collider (circular e + e − and pp) measurements of the Higgs di-photon decay rate could yield significant constraints on the values or the Higgs portal coupling needed for discovery of the H ±± H ∓ → W ± W ± hW ∓ / ± ± hW ∓ modes. For heavier triplet masses, the discovery potential for these modes would be relatively unconstrained.
The structure of this paper is as follows: in section 2, we set up the complex triplet Higgs model and discuss its key features and various model constraints. We also discuss neutrino mass generation from the type-II seesaw mechanism as well as experimental constraints on the neutrino masses. In section 3, we focus on how to determine the model parameters from future collider measurements, and in section 4, we study production cross sections and decay patterns of the triplet Higgs particles. Section 5 presents our result for model discovery at the 100 TeV collider, and section 6 discusses a strategy for the determination of λ 4 . Section 7 is our conclusion, and we summarize the details in the appendices.

The complex triplet Higgs model
In this section, we will discuss setup of the triplet model and various model constraints. We will also discuss key features of the model in section 2.3 and close this section by illustrating how neutrino masses are generated through a Type-II seesaw mechanism and by discussing current constraints on the neutrino masses.

Model setup
The type-II seesaw model contains the SM Higgs doublet Φ with hypercharge Y Φ = 1 and the complex triplet Higgs field ∆ with hypercharge Y ∆ = 2 [8] written in a matrix form [5,6,9,10] GeV, which is the scale of electroweak spontaneous symmetry breaking (EWSB). And as will be discussed below,

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v ∆ will be strongly constrained by the ρ parameter. This scalar extension of the SM is also know as the complex triplet Higgs model (CTHM).
The kinetic Lagrangian is with the covariant derivatives where g and g are the U(1) Y and SU(2) L gauge couplings, respectively. The second term in D µ ∆ introduces new interactions between the electroweak gauge bosons and the triplet, which contributes to the masses of the former when the triplet gets a nonzero vev. We write the general CTHM potential as where m and M are the mass parameters and λ i (i=1,. . ., 5) are the dimensionless quartic scalar couplings, which are all real due to hermiticity of the Lagrangian. The µ parameter, however, is in general complex and, thus, a possible source of CP violation (CPV). But as discussed in refs. [131,132], the CPV phase from µ is in fact unphysical and can always be absorbed by a redefinition of the triplet field. After EWSB, the minimization conditions imply that We will use the same notation below. The scalar states are, in general, mixtures of the field components that carry the same electric charge: (ϕ, δ, χ, η); (ϕ ± , ∆ ± ); and H ±± , which is already in its mass eigenstate. The absence of a CPV phase in the potential implies that the real and imaginary parts of the neutral doublet and triplet fields cannot mix with each other. To diagonalize the -4 -where κ = 8 or 16 depending on one's choice on the partial wave amplitude of an elastic scalar scattering from the consideration of S-matrix unitarity. For detailed discussion, see ref. [131].
• Perturbativity: keeping only the top Yukawa coupling, gauge interactions, and scalar potential couplings, the one-loop renormalization group equations (RGEs) rewritten in our notation are 4

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3 Here and below, "&" means the logical conjunction "and". 4 Two-loop RGEs for the Higgs portal parameters have been studied in ref. [21].
with t ≡ ln(µ/m t ). For perturbativity, we require a similar approximate condition on the quartic Higgs couplings as in ref. [143], which is based on the work of ref. [144] i.e., where λ FP 12 in the renormalization of ref. [145] and Λ is the cutoff scale of the theory. Figure 1 gives constraints from VS (green region) and PU (orange region) at tree-level. The black dot corresponds to our benchmark point discussed in section 6.1, i.e., After solving the above mentioned RGEs, one finds that VS and perturbativity up to the Planck scale impose stringent constraints on λ i 's [21]. For our benchmark point as input at the scale µ = m t , the resulting running couplings are shown in figure 1. From the right panel of figure 1, it is clear that the CTHM stays perturbative even at the Planck scale. We also find that the potential develops a second minimum at O(10 5 -10 6 GeV). The presence of this second minimum implies that the SM vacuum may become either unstable or metastable above this scale. In principle, stability could be preserved to higher scales with the presence of additional contributions to the RGEs associated with particles heavier than this threshold. A detailed investigation of the possible U.V. embedding of the CTHM goes beyond the scope of the present study. We observe, however, that the stability region for our benchmark point lies well above the range of triplet scalar masses that we consider below. Moreover, one may also increase the scale at which the potential may develop a second minimum by increasing λ 4 while preserving perturbativity, which is indicated by the black arrow in the left panel of figure 1. We will discuss this point further in section 5.4.

Key features of the CTHM
Since v ∆ v Φ due to the ρ parameter constraint, we expect, in general, tan 2α (and thus sin α) to be small. In this case, we have from eq. (2.12), Then in this small sin α limit, the expressions for the masses given in eq. (2.13)-(2.17) can be simplified to We see that m ∆ sets the overall mass scale of the triplet scalars whereas λ 1 is basically determined by m h and v. Moreover, in the large m ∆ limit, the mass splitting is which depends only on λ 5 , m ∆ , and v. Thus, by measuring the masses of any two triplet scalars of differing charges, one could determine both m ∆ and the Higgs portal coupling λ 5 . A practical corollary is in the large m ∆ limit, once one of the triplet Higgs particles is discovered, the relatively small mass splitting (compared to m ∆ ) would provide guidance as to the mass region for discovery of the other triplet Higgs scalars.

Neutrino masses from a type-II seesaw mechanism
In the CTHM, the neutrino masses are generated through a type-II seesaw mechanism via the Yukawa Lagrangian [5,6,9,10] Here, L = (v L , e L ) T is the l SU(2) L doublet; h ν is the neutrino Yukawa matrix, which is a 3 × 3 complex and symmetric matrix as has been shown for a general case in ref. [146].
After the EWSB with v ∆ = 0, neutrinos of different flavors mix through h ν , as implied -9 -

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by neutrino oscillations. The mass matrix h ν v ∆ also breaks the lepton number explicitly, 5 implying that neutrinos are of the Majorana type with their masses being Experimentally, sum of neutrino masses is constrained to be i m i < 0.23 eV by the Planck Collaboration via assuming the existence of three light massive neutrinos, the validity of the Λ Cold Dark Matter (ΛCDM) model and using the supernovae and the Baryon Acoustic Oscillations data [133,147]. Given this constraint, we choose m ν = 0.01 eV for each of the three light neutrinos throughout the paper. In principle, one can choose a larger (smaller) value for the neutrino masses while still satisfying the experimental constraints. Larger (smaller) neutrino masses will correspond to a larger (smaller) h ν for fixed v ∆ , which will in turn affect the same-sign di-lepton decay BRs of H ±± . The BRs will then affect the parameter space relevant for model discovery. We will discuss effects from smaller/larger m ν in section 5.4.

Model parameter determination
The model parameters for the CTHM are, naïvely, {g, g , v Φ , v ∆ , µ, λ i , h ν } (i = 1, 2, 3, 4, 5), or in the mass eigenstates after the EWSB, , G F , m Z , m h are already well-known from electroweak precision and Higgs mass measurements, and in order to further determine other parameters of the CTHM, we will need discovery of the new particles to know their masses and the measurement of the mixing angle sin α as well. Therefore, in the following sub-sections, we will discuss how to experimentally determine the other parameters of the CTHM. In the end of this section, we will also discuss how to determine the input model parameters from consideration of perturbativity, which is essential for our collider study in section 5 and section 6.
3.1 Mass spectrum and determination of λ 1 and λ 5 From section 2.3, we conclude that sin α is in general small and in this small sin α limit, we have eq. (2.41), i.e., we see that: (a) When we call this the Normal Mass Hierarchy (NMH); (b) while when λ 5 ≥ 0, m H ±± ≤ m H ± ≤ m A m H and m h < m H , we call this the Reversed Mass Hierarchy (RMH). For the NMH, SM h is the lightest particle and H ±± is the heaviest one, the order of the mass spectra is unique. While for the RMH, A or equivalently H is the heaviest particle, but the mass order between h and (H ± , H ±± ) is unclear and will generally depend on our model input. 5 In principle, one could assign a lepton number of -2 to ∆ so that the overall Lagrangian conserves lepton number before EWSB. The third term in V (Φ, ∆) would then explicitly break lepton number conservation. The coefficient of the dimension five lepton number violating mass termL C H T HL is then proportional to µ/M 2 .  On the other hand, from m 2 h 2v 2 λ 1 , we conclude that λ 1 ≈ m 2 h 2v 2 ≈ 0.129. While to determine λ 5 , one can use the mass splitting ∆m ≈ |λ 5 |v 2 8m ∆ as defined in eq. (2.42) upon discovery.

Measurement of the mixing angle sin α for determination of λ 4
To determine λ 4 , we note that from eq. (2.40), we can solve for α: which implies sin α is in general a two-to-one function. This feature of sin α is graphically reflected in figure 2. In addition, from figure 2, we see that sin α indeed decreases with increasing m ∆ and/or decreasing v ∆ . For example, when m ∆ 300 GeV and/or v ∆ 0.1 GeV, sin α 0.01.
On the other hand, the variation of sin α with λ 45 can also be used to determine λ 45 through various gauge boson-Higgs couplings. We focus on gauge boson-Higgs vertices as electroweak production of the triplet Higgs particles is the dominant production mechanism in the CTHM. After a careful investigation of all the triple vertices listed in appendix D, we find that only four of the gauge boson-Higgs couplings, as listed in table 1, are linearly dependent on sin α. 6 These couplings will eventually affect the decay BRs of the BSM particles. Thus, after their discovery, one could determine λ 5 from the mass splitting and λ 4 from the triplet Higgs decay BRs. 7 6 Some of the non gauge boson-Higgs type vertices are also sin α linearly dependent as can be seen from the hH ++ H −− vertex in appendix D, but the corresponding production cross section is smaller compared with the dominant electroweak production. 7 Here we remind the reader that the Higgs portal parameters λ4,5 are of particular interest as they may allow a SFOEWPT to explain the baryon asymmetry of the universe (BAU). In this paper, however, we will not discuss the effects on phase transition or baryogenesis from the CTHM but rather leave it for future work.

λ 2 and λ 3 determination
Different from the determination of λ 4 and λ 5 , however, λ 2 and λ 3 are in general very difficult or even impossible to measure as they are always suppressed by v 2 ∆ (for mass terms) or by v ∆ (for three-body interactions). One possible way to measure them is through the quartic triplet Higgs interactions, but the production cross section will again be suppressed by the smallness of v ∆ in general. Note that since λ 2 and λ 3 are irrelevant to electroweak phase transition, it is unnecessary to pay too much attention to their determination.

Choice of input model parameters
As discussed in last three sub-sections, experimentally, one can use the SM Higgs mass, the mass difference and the mixing angle to determine λ 1 , λ 4 and λ 5 . But recall that, in section 2.2.1, the ρ parameter requires v ∆ to be negligible compared with v Φ or v, which is about the same order as the Higgs masses. The ratio of the Higgs masses and v ∆ will then lead to very large λ 2,3 by referring back to eq. (2.21)-(2.22), thus to preserve perturbativity of the CTHM, one will have to "fine-tune" the Higgs masses to obtain reasonable values for λ 2,3 . To avoid the "fine tuning", we choose λ 2,3 instead as our input in our theoretical study. As also discussed in section 3.1 and section 3.2, (a) Since we know the Higgs mass exactly, we choose m h instead of λ 1 as our model input; (b) we choose m ∆ and λ 5 as our model input as they determine the mass spectrum; (c) sin α is negligible at small v ∆ , thus to avoid "fine tuning" λ 4 , we choose λ 4 instead of sin α as our model input. Another reason for choosing λ 4 as our model input is that it frequently always appears in pair with λ 5 such that one can infer λ 4 from the combination once we know λ 5 . At the same time, relevant quantities may depend separately on λ 4 and λ 5 , e.g., H ± decay BRs. To summarize, our Here we emphasize that the input parameters need to be carefully chosen to avoid fine tuning the masses or to preserve the validity of perturbation theory from λ 2,3 , otherwise one may easily fall into the region where perturbation theory is invalid. For example, for the plots in the second row of figure 2 in ref. [72], the authors used the scalar masses as their input. We find that using their input, only when v ∆ 1 GeV will the value of λ 3 respect perturbativity, whereas for smaller v ∆ 's, λ 3 can be as large as 10 21 .

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4 Production and decay rates of the scalars in the CTHM As discussed in last section, the mass ordering of the RMH will in general depend on our model input. For simplicity, we will work in the NMH throughout the paper, in which framework the production and decay rates of the BSM Higgs particles are studied in detail below. While we want to point out that, in the RMH, though the decay patterns, the decay BRs and thus our figure 7 and figure 11 will change, the same channels studied in this paper can still be used for model discovery and Higgs portal parameter determination.

Production cross section of the Higgs particles in the CTHM
In SM, the Higgs boson can be produced via gluon fusion or vector boson fusion (VBF), but in the CTHM, single production of the triplet Higgs particles via gluon fusion or VBF is highly suppressed by small v ∆ . 8 Therefore, single production of the triplet Higgs particles through gluon fusion or VBF will not be considered in this paper. For double scalar production, a pair of triplet scalars can be produced through electoweak Drell-Yan processes or gluon fusion. As in the single Higgs production case, however, double scalar Higgs particle production via an intermediate H or A, which is produced through gluon fusion, is again highly suppressed by small v ∆ . No such suppression occurs for electroweak pair production. Consequently, we focus on the latter.
To study quantitively the production cross sections of the triplet Higgs particles, we first use Mathematica and FeynRules 2.3.13 [148,149] to generate the Universal FeynRules Output (UFO) model file [150] of the CTHM, then we use MadGraph 2.3.3 [151] to implement the CTHM UFO file to obtain the production cross sections at √ s = 14 TeV and √ s = 100 TeV. However, we find that for the channels we are going to study in this paper, the number of events at √ s = 14 TeV and L = 3 ab −1 is too few even without considering the corresponding backgrounds, so we only list the cross section result at √ s = 100 TeV here. The pair production cross sections depend on the couplings of the electroweak gauge bosons to the scalars and on the scalar masses. In what follows, we cast these dependences in terms of our independent parameters. Note that λ 1 is basically fixed by v and m h , while the effects of λ 2,3 are suppressed by small v ∆ . In short, the production cross sections will be largely insensitive to λ 2,3 but will depend significantly on λ 4,5 . To be consistent with the NMH, which requires a negative λ 5 , and to satisfy the constraints discussed in section 2.2, we choose λ 2 = 0.2, λ 3 = 0, λ 4 = 0 and λ 5 = −0.1. As an example, we fix v ∆ = 10 −3 GeV and obtain the production cross sections given in figure 3, from which we see that pair production of H ++ H −− has the largest production cross section followed by H ++ H − . On the other hand, H + H −− will always be produced simultaneously with H − H ++ . We therefore expect an enhancement of the cross section from the combination of H − H ++ and H + H −− channels.
The hierarchy of the various production cross sections is briefly explained below: (a) Besides a factor of four enhancement from the electric charge of H ±± , H ++ H −− pair has a larger cross section than H + H − because it is constructively produced through s-channel γ and Z exchange. In contrast, the H + H − pair production is suppressed due to destructive 8 SM h production via gluon fusion, however, does not suffer from suppression from the smallness of v∆.

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���������� ����� ������� �� � =������ Figure 3. Production cross section as a function of m ∆ at √ s = 100 TeV with v ∆ = 10 −3 GeV. We set λ 2 = 0.2, λ 3 = 0, λ 4 = 0 and λ 5 = −0.1, which correspond to the black dot in the left panel of figure 1 in order to be consistent with the NMH framework and to satisfy the model constraints discussed in section 2.2.2. The left panel is for associated Higgs production channels while the right one is for pair production except the HA channel. Since the production cross section of HA is very close to H − H ++ , we include it in the right panel to make the plots more readable.
interference [57]. Note that even though m H ±± > m H ± , the mass splitting is not large due to our choice of λ 5 ; therefore, the lighter H ± mass does not compensate for the aforementioned factors. (b) H ++ H − has a larger cross section than H −− H + because the former is dominantly produced through a W + while the latter is through a W − . (c) HH and AA channels, or H ± A and H ± H channels, have the same production cross sections due to mass degeneracy of H and A. (d) H ± A/H ± H has a smaller cross section than HH/AA, and HA has a smaller cross section than H ++ H −− /H ++ H − , because of the couplings. (e) In the NMH, m H ± > m H/A , but the couplings involved for H + H − is larger than those for H + A/H + H, the phase space and the couplings will compete such that at small m ∆ , H + H − has larger cross section while at large m ∆ , H + A/H + H has a larger cross section. This is also true for HA and H + H − channels.
In order to study the collider signatures of the triplet Higgs particles, it is natural to focus on H ±± H ∓∓ and H ±± H ∓ channels since they have the largest production cross sections compared with other channels. To determine the final states, we will study their dominant decay channels in next sub-section.

Decay rates of the scalar Higgs particles in the CTHM
To further determine the dominant decay modes of the triplet Higgs particles in the CTHM for collider simulation, we calculate their decay rates by taking h ν = 3×3 for simplicity. All our decay formulas agree with those in appendix A of ref. [72] if one also takes the unit matrix limit there.
In order to illustrate the potential parameter-dependence of various decay channels, we show in figure 4 the BRs for the charged and neutral triplet states as functions of the relevant combinations of λ 4 and λ 5 for representative values of m ∆ = 400 GeV and v ∆ = 10 −4 GeV. In this study, we will focus on the NMH with λ 5 < 0. From the top left panel of figure 4, we observe that the H ±± BRs to H ± W ± and W ± W ± depend strongly on this parameter in the vicinity of our benchmark point value: λ 5 = −0.1. From the top right plot, we also observe that the BR(H ± → hW ± ) also depends strongly on λ 4 + λ 5 . Even though in the vicinity of our benchmark point with λ 4 + λ 5 = −0.1 the hW ± mode is subdominant, the corresponding BR depends more strongly on λ 4 + λ 5 than do the other modes. Consequently, we will focus on this channel for the decay of the singly-charged scalar. The bottom two panels give the neutral scalar BRs. Though we will not utilize this information in the present study, we include them here for completeness and for future reference.
It is also useful to determine how the H ±± BRs vary with m ∆ and v ∆ . To that end, in figure 5, we show the regions of parameter space where the BR to various final states is greater than 40% for H ±± . In the left panel of figure 5, we consider the (v ∆ , λ 5 ) plane for fixed m ∆ , while the right panel gives the (m ∆ , λ 5 ) plane for fixed v ∆ . Note that H ±± decay BRs are independent on λ 4 and for the NMH, one has λ 5 < 0.
From figure 5, we observe that for H ±± , the dominant decay channels are H ±± → ± ± (W ± W ± ) at small (large) v ∆ when m ∆ = 400 GeV. For intermediate values of the triplet vev, e.g. v ∆ = 10 −4 GeV, those two channels dominate when λ 5 ≥ −0.2. Besides the large BRs in the corresponding regions of v ∆ , additional advantages for these channels are: Figure 5. Decay region plots for H ±± with BR≥ 40%. Left panel is with m ∆ = 400 GeV and right panel is with v ∆ = 10 −4 GeV. Purple region is the H ± W ± channel, black is the same-sign di-W boson channel and blue is the same-sign di-lepton channel. λ 5 is in the negative region to be consistent with the NMH framework.
(1) Clean final states: leptons in the final states are relatively easy to identify and analyze experimentally; (2) Absence of cascade decay: the H ± W ± decay mode will introduce extra decay chains, making the final state more complicated. We emphasize, however, that even though the same-sign di-W boson (di-lepton) channel dominates for large (small) v ∆ , one may still probe the intermediate v ∆ region using the ± ± and W ± W ± channels. Although these channels have relatively small BRs in this v ∆ region, we find that by combining thesechannels with information from other triplet Higgses, one could still explore this region without resorting to the H ±± → W ± H ± channel. This feature will become more apparent in our main discovery reach plot figure 7 and attendant discussion.
We also note in passing that at small v ∆ , same-sign di-lepton channel dominates and actually has a 100% decay BR. For those regions where the same-sign di-lepton channel has a 100% decay BR, experimental constraints are strong. We will discuss this point in detail in section 5.4.
In figure 6, we show the regions of parameter space where the H ± decay BR to various final states is greater than 40%. Since the BR functions for H ± depend on v ∆ , m ∆ , λ 4 and λ 5 individually, the decay region plots for H ± are more complicated than those for the doubly charged scalars. We thus plot the dominant decay channels in different planes: in the first row of figure 6, we consider the (v ∆ , m ∆ ) plane with varying λ 45 , while in the second (third) row, we consider the (v ∆ , λ 5(4) ) plane with fixed λ 4 (5) and v ∆ . Recall that from table 1, only the H ± → hW ± channel is related to the determination of λ 4 through the mixing angle sin α as discussed in section 3.2. We observe that λ 45 < 0 generally leads to a large BR for the H ± → hW ± channel, though there also exist some regions giving a large BR(H ± → hW ± ) for λ 45 > 0.
With the foregoing observations in mind, we will next study the following channels for model discovery:

Present experimental constraints
Present experimental constraints on the charged Higgs particles we study here already exclude some portions of the CTHM parameter space especially from studies on the pp → H ++ H −− → + − − − ( = e, µ) process. Thus, before moving to the detailed collider study of some specific channels, we review the current direct LHC experimental constraints. A detailed summary can be found in appendix A, with the most stringent ones given below: 1. For H ±± : by assuming a 100% di-lepton decay BR, the lower limit on m H ±± is constrained to be 870 GeV [130] for a µ ± µ ± final state. In ref. [124], an upper limit on the cross section with the ± ± ( = e, µ) final state is set to be between 1.7 fb and 67 fb. While by assuming H ±± is long-lived, 9 m H ±± ∈ [50, 600] GeV is excluded [128].
GeV [153]. Here, a larger mass corresponds to a smaller upper bound on the product of the production cross section and the BR. A similar meaning is implied in the following.
3. For H and A: in ref. [154], the upper limit on For the charged Higgs particles, we will recast constraints from the charged Higgs particles to the parameter space of the CTHM in section 5.4, in which we show the part of the parameter space that is already ruled out by current experimental constraints for the benchmark point we choose.

Model discovery
As discussed in last section, H ++ H −− has the largest production cross section and will be the dominant discovery channel for the triplet model; H ± H ∓∓ has the second largest production cross section and is directly related to the determination of λ 4,5 . In addition, since the same-sign di-lepton decay channel of the H ±± particle is dominant only at small v ∆ from left panel of figure 5 and the H ± → hW ± decay channel dominates at large v ∆ from first row of figure 6, we expect these two channels to be complementary to each other to cover most of the model parameter space. Therefore, in this section, we will study in detail the discovery of the triplet model through these two channels, i.e., pp → H ++ H −− and pp → H ±± H ∓ with H ±± → ± ± /W ± W ± and H ± → hW ± . 9 As explained in the footnote of ref. [128], "long-lived" means a particle that does not decay within the full depth of the ATLAS detector. Figure 6. Decay region plots for H ± with BR ≥ 40%. Purple region is for HW and AW , blue for ZW , orange for hW and black for the lepton final state. The first row is with the same λ 5 but opposite-sign λ 4 ; the second row is with the same v ∆ but opposite-sign λ 45 and the third row is with the same v ∆ but different λ 5 . From those plots we conclude that H ± → hW ± channel prefers λ 45 < 0 in general. For λ 5 = −0.01, H ± → hW ± also gains a large branching ratio when λ 4 goes from negative to positive as can be seen from the last graph.   (1) at a lepton collider with √ s = 380 GeV and 3 TeV, the production and decays of H ±± were studied by Agrawal et al. [117]; (2) the H ++ H −− pair production cross section at the future 100 TeV pp collider was studied by Cai et al. [118]; (3) the H ++ H −− → τ ± ± ∓ ∓ / + τ + − τ − processes were studied by Li [119] at the high-luminosity and highenergy LHC as well as the future 100 TeV circular pp collider (FCC); (4) the multi-lepton final state of H ++ H −− at 13 TeV LHC and FCC was studied by Mitra et al. [54] in the RMH by fixing λ 1 = 0.13 and λ 2 = λ 3 = λ 4 = 1. To the best of our knowledge, in the NMH this channel at the FCC has not yet been studied.
In what follows, we discuss our collider simulation for this channel with a mass range from 40 GeV to 5000 GeV. The simulation is done by using MadGraph 2.3.3 [151] and the aforementioned pre-generated CTHM UFO file to generate events, and then each generated event undergoes parton shower and hadronization through Pythia-pgs 2.4.4 [155] before arriving at the detector. The detector response is simulated by Delphes 3.3.0 [156], where the 100 TeV FCC Delphes card [157] is used at this step. To analyze the data collected by Delphes, we use ROOT 6.06.02 [158].
The dominant backgrounds for this channel are ZW ± W ∓ and ZZ as we are performing an exclusive analysis. In total, we generate 1,000,000 events for both the signal and the two backgrounds, and our preselection cuts for the signal and the backgrounds are: (1) transverse momentum p T > 20 GeV for all final state particles; (2) absolute pseudorapidity |η| < 2.5 for all final state particles. Since the Boosted Decision Trees (BDT) [159] can maximize the cut efficiency and thus have better performance than a cut-based analysis [160], we will utilize this feature of BDT to train and test all the events that have passed the preselection cuts. We list the variables used during BDT training and test in table 2. Table 3. Signals for intermediate and large v ∆ are listed in the first two rows. The two signals share the same backgrounds, which are listed in the following eight rows.

Discovery for large
From the BR discussion in section 4.2, we observe that the H ++ H −− → + + − − channel can only cover the small v ∆ region, and we expect the large v ∆ region to be covered by In this paper, we only focus on the W ± → ± ν mode for all the four W bosons. In this case, the 4W channel has exactly the same backgrounds as the H ++ H −− → + + − − channel considered in last sub-section.
Repeating the same procedures as for the H ++ H −− → + + − − channel, we generate 1,000,000 events for our signal and use the background data generated in last sub-section. We also use the same BDT training and test variables as those listed in table 2 to analyze this channel.

Discovery for intermediate and large
only covers the small (large) v ∆ region, the H ± H ∓∓ can provide complementary discovery potential for the large and intermediate v ∆ region. To obtain information about λ 4,5 , we require H ± to decay into a hW ± final state, while H ∓∓ can decay into either an ∓ ∓ or a W ∓ W ∓ final state. These two processes yield the same final state particles and, thus, share the same backgrounds. The backgrounds we consider for these two processes are: hZW ± ; ttj and W ± W ∓ bbj with the light jet j misidentified as a lepton with a fake rate of 0.01% [157]; ttW ± , ttZ and ZZh with one lepton missing; ZW ± jj with the two light jets misidentified as two b quarks with a fake rate of 10% for c misidentified as b and a 0.01% fake rate for other light quarks [157]; and ZW ± bb. The signals and the backgrounds are summarized below in table 3.
As for the H ++ H −− process, we use the same tools to generate events, the same preselection cuts to analyze the events. For the BDT training and test, the training variables we use for these two processes and the backgrounds are listed in table 4. In addition, : pseudo-rapidity of the b quark with leading and sub-leading p T respectively p leading T, same , p sub-leading T, same : leading and sub-leading transverse momentum of the same-sign leptons η leading , η sub-leading : pseudo-rapidity of the same-sign leptons with leading and sub-leading p T respectively η oppo. , p T, oppo. : pseudo-rapidity and transverse momentum of the opposite-sign lepton Table 4. A list of BDT variables for W ± W ± hW ∓ , ± ± hW ∓ channels and their backgrounds. Since these two signals share the same backgrounds, we use the same BDT variables for both channels.

Discovery potential at the 100 TeV collider
The significance is defined as S √ S+B throughout the paper, with S = σ s · L and B = σ tot bkg · L the total signal and background event number at the collider, where σ s and σ tot bkg are the final signal and final total background cross section respectively, and L is the integrated luminosity, which we choose to be 30 ab −1 [161, 162] throughout the paper. By requiring the signal significance to be greater or equal to 5, the BDT based result for the discovery channels is given in figure 7. Several features of these results merit emphasizing: • We see that at small v ∆ where the neutrino masses are naturally generated through the type-II seesaw mechanism, the CTHM can be discovered over a very wide mass range from tens of GeV to several TeV through the pp → H ++ H −− → + + − − channel. We also recast the current LHC constraints for this channel at 8 TeV and 13 TeV, which is done by rescaling the production cross sections and the BRs in refs. [127,130]. We find that the current LHC constraints only exclude the relatively small m ∆ and small v ∆ region of the CTHM parameter space for our benchmark point, which therefore motivates a future collider study as we have done above.
• For the benchmark point we use, such that LEP constraints [163,164] are automatically satisfied. Note that our figure 7 is plotted as a function of m ∆ such that m ∆ = 0 corresponds a minimal mass of m H ±± 54.78 GeV.  • For the large v ∆ region, the pp → H ±± H ∓ → W ± W ± hW ∓ channel allows discovery of the CTHM up to about 1 TeV. The LHC constraints for this channel are currently absent, and the corresponding parameter space will be covered by the future 100 TeV collider. In addition, for intermediate v ∆ 's, the overlap among W ± W ± hW ∓ , ± ± hW ∓ and H ++ H −− channels can also allow us to roughly determine m ∆ ∈ [400, 1000] GeV and v ∆ ∈ [10 −4.4 , 10 −3.9 ] GeV if all these three channels are observed with significance 5 or more. The redundancy among these models would provide an important cross check that the signals are due to the CTHM.

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• For large v ∆ and large m ∆ region where the H ±± → W ± W ± channel dominates as can be seen from left panel of figure 5, one would expect the H ++ H −− → W + W + W − W − → + + − − / E T channel to cover much of that parameter space.

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Although our present analysis is not optimized to extend beyond m ∆ ∼ 1.6 TeV for this channel, one might expect use of other W decay modes (and a correspondingly different BDT training) to allow extension to higher masses. As an example, we note that the authors in ref. [107] have studied the channel pp → H ++ (→ W + ν )H −− (→ W − jj) and concluded that H ±± could be discovered at the 14 TeV LHC with L = 10-30 fb −1 . It is worth exploring whether use of this channel (or others) may also afford greater coverage for m ∆ > ∼ 1.6 TeV.
• One may also consider using the H ±± H ∓∓ → W ± W ± ∓ ∓ channel to cover part of the parameter space. We note, however, that since the same-sign di-W and the samesign di-lepton decay channels are dominant only at large and small v ∆ respectively (as can be seen from the left panel of figure 5), we thus expect these channels to have enough significance only at v ∆ ∼ (10 −5 , 10 −4 ) GeV. The same region is already well covered by the ± ± hW ∓ and H ++ H −− → ± ± ∓ ∓ channels.
• The H ++ H −− channel covers a very wide range over m ∆ at small v ∆ and the W ± W ± hW ∓ channel disappears around m ∆ =1 TeV. The reason for the "long tail" of the H ++ H −− channel can be understood from the blue region in figure 8 (a), from which we see that the BR(H ±± → ± ± ) decreases slowly with increasing m ∆ for v ∆ 10 −4 GeV, leading to a very slowly decreasing significance. In contrast, for the W ± W ± hW ∓ channel, the significance drops dramatically at m ∆ ≈1 TeV because of phase space suppression for heavier particles and decay BR suppression at smaller v ∆ 's as can be seen from figure 8(b).
• We remind the reader that we choose m ν = 0.01 eV for all the three light neutrinos generation throughout the paper. Since a larger (smaller) m ν will correspond to a larger (smaller) Yukawa coupling and thus a larger (smaller) same-sign di-lepton decay BR of H ±± , we therefore expect the same-sign di-lepton decay regions in figure 7 will shift upward (downward) for larger (smaller) m ν 's.
• Finally, for our benchmark point, vacuum stability is not guaranteed at the Planck scale as discussed in section 2.2.2. In ref. [21], it was shown that vacuum stability up to the Planck scale actually prefers positive λ 4 's as indicated by the black arrow in the left panel of figure 1. This difference is not, in general, problematic, as the stability region for our benchmark point amply covers the triplet mass range considered here. One could anticipate additional degrees of freedom modifying the behavior of the potential at larger scale, so as to ensure stability to the Planck scale. Nevertheless, it is interesting to ask how the reach indicated in figure 7 would evolve as we move along the black arrow in figure 1 corresponding to higher stability scales. We expect the discovery regions including the H ± → hW ± channel in figure 7 to shrink for 0 λ 4 3 as the H ± → hW ± decay BR decreases for λ 4 's in this region as can be seen directly from the upper right panel of figure 4. For λ 4 6, one would expect the discovery regions including the H ± → hW ± chain to expand even though one needs to re-consider all the model constraints discussed in section 2.2. For these larger values of the λ 4 , however, we would expect to reach the limit of perturbativity well below the stability scale.
Product of the BRs is suppressed for small v ∆ 's, which explains feature of the W ± W ± hW ∓ channel in figure 7 in the small v ∆ region.

Triplet Higgs potential determination and simulation
From our result in the previous section, for m ∆ 4500 GeV, the H ++ H −− → + + − − , W ± W ± hW ∓ and ± ± hW ∓ channels can cover a significant portion of the parameter space of the CTHM except the region where m ∆ 1 TeV and v ∆ 10 −4 GeV. We expect some of the latter region to be covered by the H ++ H −− → W + W + W − W − channel as discussed in last section. Therefore, the discovery potential for the CTHM at a 100 TeV pp collider is considerable. Assuming discovery of the doubly-and singly-charged scalars, we can fix λ 5 straightforwardly through the mass splitting as discussed in section 3.1. However, to determine the important Higgs portal parameter λ 4 , additional information will be needed.
For v ∆ larger than ∼ 10 −5 GeV, the BR for H ± → hW ± is particularly useful as discussed in section 3.2. 10 To investigate this possibility, we adopt the following strategy. First, we will carry out a detailed simulation for a choice of λ 4 + λ 5 in the region where the BR(H ± → hW ± ) is strongly-dependent on λ 4 + λ 5 , according to the top right panel of figure 4. We will carry out this study for a choice of the λ j consistent with the stability and perturbativity considerations discussed above and for two different choices of the overall triplet mass scale, m ∆ . Second, we will scan over the values of λ 4 and m ∆ for fixed λ 5 , thereby varying JHEP01(2019)101 the production cross section and BR from the values corresponding to our benchmark points. In doing so, we will rescale the significance of the signal accordingly. Third, we will repeat this analysis for different representative choices of v ∆ to indicate how the varying H ±± BR affects the λ 4 -sensitivity. Finally, we will compare the sensitivity with that of the observation of the rate for the SM Higgs boson to decay to a di-photon pair, as loop corrections from charged triplet scalars will affect the corresponding rate as functions of the Higgs portal couplings and m ∆ . The results are plotted in figure 11, where we show the corresponding regions of 5σ sensitivity to the model parameters.
In what follows, we provide a more detailed discussion of the collider simulation and analysis than we provided for the results in figure 7, given that we focus on the H ± → hW ± channel for coupling determination.

Benchmark points
As discussed in section 4.2, the H ++ H −− → + + − − channel is powerful for the triplet model discovery at small v ∆ , but it can not determine λ 4 as it is λ 4 -independent. In contrast, H ∓ H ±± → hW ∓ ± ± /hW ∓ W ± W ± are promising for the determination of λ 4 at intermediate and large v ∆ . In order to determine their collider signatures, we choose two representative benchmark points, taking into account vacuum stability, perturbative unitarity, perturbativity, neutrino masses and our result in figure 7: m ∆ = 800 GeV (m ∆ = 400 GeV), v ∆ = 10 −4 GeV, m h = 125 GeV, m Z = 91.1876 GeV, m ν = 0.01 eV, λ 2 = 0.2, λ 3 = 0, λ 4 = 0, λ 5 = −0.1 for the W ± W ± hW ∓ ( ± ± hW ∓ ) channel, which is a representative point of the large (small) m ∆ region. Note that although these benchmark parameter choices have λ 4 = 0, the sum λ 4 + λ 5 differs from zero and lies in a region where BR(H ± → hW ± ) varies significantly with this combination of couplings. The choice of two the two different mass scales corresponds to the edges of various overlapping discovery regions, as indicated by the two black points in figure 7.

Simulation
In this section we will first generate data for pp → H ∓ H ±± → hW ∓ ± ± → bb ∓ ± ± / E T using MadGraph, and then analyze the data by both a cut-based analysis and using the BDT method. In the former, we choose a set of "hard cuts" by first comparing various signal and background distributions and endeavoring to optimize by hand the choice for greatest signal significance. As an alternative, we employ the BDT. As we show below, the BDT method generally provides a better signal efficiency and significance.

Cut based analysis: basic cuts
While analyzing the data by ROOT 6.06.02, we require all the final state particles have transverse momentum p T > 20 GeV and pseudorapidity |η| < 2.5; we also require exactly three leptons in the final state 11 and exactly two jets in the final state 12 for the signal and the ttW ± , ttZ, hZW ± , ZW ± bb and hZZ backgrounds.
Events # (normalized)  Figure 9. Representative reconstructed variables for the ± ± hW ∓ channel after the basic cuts. We use the word "signal" to represent the pp → H ±± H ∓ → ± ± hW ∓ channel in all histograms above.
backgrounds, we require there are exactly two leptons and three jets 13 in the final state.
For the ZW ± jj background, when the light jet is a c quark, we use a fake rate of 10%; and when the light jets are other light quarks, we use a fake rate of 0.01% [157]. After the basic cuts, the result of reconstructed variables is given in figure 9, and the cut efficiency is given in table 5. By comparing the signal and the background distributions in figure 9, we find that ∆φ and ∆R between the two b quarks, scalar sum of the transverse 13 With at least one and at most two of the three jets are b quarks. The light jet with the smallest pT among these three jets is taken to be a lepton with a 0.01% fake rate [ Table 5. Cut flow table for pp → H ±± H ∓ → ± ± hW ∓ under basic cuts (bc) and hard cuts (hc) with integrated luminosity of 30 ab −1 . Here and in table 9, we use the same abbreviations: "proc." for "processes"; "E" for "base 10 exponential function"; "cs" for "cross section" with unit f b; "eff." for "efficiency" in percent; "signi." for "significance" and "hci-j" means "applying hard cuts i, · · · , j".
momentum H T , same-sign lepton leading and sub-leading p T , same-sign lepton ∆φ and ∆R, m h , m H ±± and W boson transverse mass m W T have distinct features between our signal and the backgrounds, which can be exploited to reduce the backgrounds. These variables are the hard cuts we apply next.

Cut based analysis: hard cuts
To improve the significance of the signal, we apply the following hard cuts in the same order as they are listed in table 6. After applying them, the cut efficiency for each hard cut and significance of our signal are presented in table 5. From the table, it is seen that the backgrounds are efficiently reduced and our signal has a final cross section about 7.3848 × 10 −4 fb, with the significance being around 4; and the estimated event number for the signal after the basic cuts and the hard cuts is around 22 at the FCC with L = 30ab −1 .

BDT based analysis result
To improve the cut efficiency, we also carry out a BDT based analysis as for analyzing model discovery at the 100 TeV collider in section 5. The result is shown in parallel with the cut-based result in table 7 for comparison, and we find that the BDT method improves the signal significance by about a factor of 2 through optimizing the cut efficiency; in    addition, the signal efficiency as well as the signal cross section are also improved by about a factor of 3.

Simulation:
The H ∓ H ±± → hW ∓ ± ± channel is helpful for the determination of λ 4 only at intermediate v ∆ , for large v ∆ 's, the H ∓ H ±± → hW ∓ W ± W ± channel can be used. Since it shares the same backgrounds as the H ∓ H ±± → hW ∓ ± ± channel in last sub-section, we generate 1,000,000 events for this signal and use the background data generated for the H ∓ H ±± → hW ∓ ± ± channel to study its collider phenomenologies. We still perform an exclusive analysis, and by using the same basic cuts as for the ± ± hW ∓ channel, we obtain the reconstructed variables under basic cuts for the W ± W ± hW ∓ channel shown in figure 10. Note that ∆Φ and ∆R between the two b quarks and the two same-sign leptons, leading p T of the same-sign leptons, SM h, the doubly-charged Higgs and Z boson masses and the transverse W boson mass are the hard cuts that can be applied to further separate the signal from the backgrounds. Those hard cuts are applied in the same order as they are listed in table 8.   Figure 10. Reconstructed variables for the W ± W ± hW ∓ channel under basic cuts. We use the word "signal" to represent the pp → H ±± H ∓ → W ± W ± hW ∓ channel in all histograms above.
Results after applying the hard cuts are given in table 9. And for comparison, the BDT based analysis is presented in parallel in table 10, we see that BDT based analysis still gives a larger significance, which is about three times larger compared with cut-based result.
6.4 Determination of λ 4 upon discovery at the future 100 TeV collider As we have been addressing throughout the paper, the H ∓ H ±± → hW ∓ ± ± and the H ∓ H ±± → hW ∓ W ± W ± channels are important for the determination of λ 4 , but our study above is done at only one benchmark point for both H ∓ H ±± → hW ∓ ± ± and H ∓ H ±± → hW ∓ W ± W ± . To see how our result is sensitive to λ 4 , we fix λ 5 = −0.  perform a scan in the λ 4 -m ∆ plane. 14 Doing so, it is straightforward to rescale the signal and, thereby, obtain the variation in signal significance. The corresponding results are given in figure 11 (a), (b), (c) with v ∆ = 10 −1 GeV, v ∆ = 10 −4 GeV and v ∆ = 10 −5 GeV respectively. There, we indicate the regions giving larger than 5σ significance for the two channels considered here.
In figure 11(a), i.e., at large v ∆ = 10 −1 GeV, only the W ± W ± hW ∓ channel is useful, whereas the significance for ± ± hW ∓ is less than 5 in the entire parameter space. The reason is that the rate for H ±± → ± ± is highly suppressed at large v ∆ as can be seen from left panel of figure 5. For W ± W ± hW ∓ , the appearance of the region at the upper-left 14 Note that λ2,3 are suppressed by v∆, so their values do not matter here.  Figure 11. Blue is significance ≥ 5 region for the hW ∓ W ± W ± channel and magenta is that for the hW ∓ ± ± channel. The outermost very light black region is the combined constraint on R hγγ from ATLAS and CMS at 7 TeV and 8 TeV; the intermediate light black region is the planned FCC-ee constraint and the innermost black region is the planned FCC-ee+FCC-hh constraint on R hγγ .
corner is due to an increase of the decay BR for H ± → hW ± when λ 4 goes from negative to positive as can be seen from the upper right panel of figure 4. Therefore, at large v ∆ , the W ± W ± hW ∓ channel is more helpful for the determination of λ 4 at the FCC. From figure 11(b), i.e., corresponding to intermediate v ∆ = 10 −4 GeV, both W ± W ± hW ∓ and ± ± hW ∓ can help to determine λ 4 . The W ± W ± hW ∓ channel covers a larger region at a higher mass scale while the ± ± hW ∓ channel provides more coverage at a lower mass scale. The overlap between these two channels makes them useful as a cross check if the triplet scale is around m ∆ ∈ [400, 900] GeV. For m ∆ ∈ [900, 1100] GeV, the W ± W ± hW ∓ channel can be used to determine λ 4 ; and for m ∆ ∈ [300, 400] GeV, we can use the ± ± hW ∓ channel.
And from figure 11(c), i.e., at small v ∆ = 10 −5 GeV, only the ± ± hW ∓ channel can be used to determine λ 4 since the H ±± → W ± W ± channel is highly suppressed as can be seen from the left panel of figure 5. Comparing this result with those at v ∆ = 10 −1 GeV and v ∆ = 10 −4 GeV, we see that at v ∆ = 10 −5 GeV, the ± ± hW ∓ channel covers the largest mass region up to about 1.4 TeV.
It is now interesting to consider the possible complementarity between these direct probes of the Higgs portal coupling and mass with indirect tests. As has been studied in refs. [71,165], the doubly-charged Higgs particle of the CTHM can give a sizable contribution to the h → γγ decay rate especially for negative λ 4 and λ 45 due to a constructive interference [23]. We therefore expect the h → γγ decay rate to provide an indirect determination of λ 4 by excluding some of the parameter space on the λ 4 -m ∆ plane. In this context, we consider the ratio R hγγ given with Γ NP and Γ SM the new physics (NP) and pure SM contribution to the decay rate of h → γγ respectively. From eq. (6.1) we see that, if nature is completely described by SM, then this ratio will exactly be one; and any value that deviates from one might be a source of new physics. For the quark loop contributions, we retain only the dominant t -31 -

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quark for the fermion loop contribution to R hγγ . The current LHC and the proposed FCC constraints on this ratio is indicated in the λ 4 -m ∆ plane in figure 11 (a), (b), (c), 15 where the lightest black region is the combined constraint on R hγγ from ATLAS and CMS at 7 TeV and 8 TeV; the intermediate black region is the planned FCC-ee constraint and the darkest black region shows the combined planned FCC-ee+FCC-hh constraint on R hγγ . From figure 11 (a), we see that the current LHC constraint on R hγγ is almost ruled out the small m ∆ and large λ 4 region, but in other regions, the current LHC constraints on the λ 4 -m ∆ plane are relatively weak. This situation, however, will be changed considerably by the future 100 TeV collider as can be seen from the darker black region in figure 11 (a), (b), (c).
Thus, combination of the direct and indirect probes of the CTHM would be advantageous in the determination of λ 4 . If future precision measurements of the h → γγ decay rate agree with the SM expectations, a substantial portion of the λ 4 -m ∆ parameter space will be excluded, thereby assisting in the determination of λ 4 . In the remaining regions of parameter space, λ 4 could eventually be determined by H ∓ H ±± → ± ± hW ∓ and H ∓ H ±± → W ± W ± hW ∓ based on our study above. It is also possible that future experiments at the LHC, FCC-ee, or FCC-hh see a deviation of R hγγ from the SM prediction. In this case, if λ 5 is determined from mass splitting (-0.1 in our case), we might also conclude that: (1) If the deviation is detected through the hW ∓ W ± W ± (hW ∓ ± ± ) channel, the triplet will have a large (small) vev with |λ 4 | ∼ 1; (2) if the deviation is observed from both hW ∓ W ± W ± and hW ∓ ± ± channels, an intermediate triplet vev can be inferred with |λ 4 | ∼ 1.

Conclusion
In this paper, we have investigated the model discovery and Higgs portal parameter determination of the Complex Triplet Higgs Model at a prospective 100 TeV pp collider. The triplet with Y=2 has long been known as a key ingredient in generating non-zero neutrino masses through the type-II seesaw mechanism. The triplet interacts with the SM through its electroweak gauge interactions, its coupling to the leptons in the type-II see saw interaction, and to the Higgs doublet via the Higgs portal parameters λ 4 and λ 5 . The latter modify the scalar potential and may enable a strong first order electroweak phase transition, as needed for electroweak baryogenesis.
The CTHM parameter space is constrained by current experiments at the LHC in the region where the triplet is light ( 600 GeV) and its vev, v ∆ , is small ( 10 −4.6 GeV). In this paper, we have analyzed the reach of a prospective 100 TeV pp collider by working in the Normal Mass Hierarchy (NMH) framework, wherein the doubly-charged Higgs particle H ±± is the heaviest. Based on our study, we conclude that a large part of the CTHM parameter space will be covered by the 100 TeV collider in the future as shown in our figure 7. More specifically, we find that: 1. The H ++ H −− and H ±± H ∓ channels have the largest and the second largest cross section respectively, making them the dominant discovery channels of the CTHM.

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Importantly, the H ++ H −− → + + − − channel is recognized as the smoking-gun signature of the CTHM, which can be used to discover the triplet up to a mass ∼4.5 TeV when v ∆ 10 −4 GeV. In addition, for v ∆ 10 −4 GeV, the triplet model can be discovered by the H ±± H ∓ → ± ± hW ∓ /W ± W ± hW ∓ channel when the triplet mass is below ∼1 TeV.

2.
For v ∆ 10 −4 GeV, the triplet can also be discovered through the H ++ H −− → W + W + W − W − → + + − − / E T channel when the triplet mass is below ∼1.7 TeV. In arriving at this conclusion, we use the same BDT training and test variables as for the H ++ H −− → + + − − channel. However, if one were to choose a different set of BDT training and test variables to optimize the cut efficiency, or if one were to study different final states like in ref. [107], one might anticipate that the quartic-W channel will also cover the upper right white corner in figure 7, such that the whole parameter space can be explored at the future 100 TeV collider.
3. Upon discovery, Higgs portal parameter λ 5 can be determined straightforwardly from the mass splitting ∆m ≈ |λ 5 |v 2 8m ∆ defined in eq. (2.42). While the triplet can be discovered over a wide range and λ 5 can be calculated straightforwardly from the mass splitting upon discovery, determination of the other Higgs portal parameter λ 4 is more complicated even after discovery. Fortunately, we can obtain λ 4 through precise measurements of the decay branching ratios. We find that only four decay vertices are helpful and summarize them in table 1. At the same time, to further narrow down the parameter space, precise measurements on the h → γγ decay rate can help indirectly to the determination of λ 4 by excluding some of the parameter space, as shown in our figure 11.
In this work, we only focus on the charged triplet Higgs particles in the NMH framework. However, the neutral triplet Higgs particles can also be used for model discovery and the Higgs portal parameter determination at the 100 TeV collider. Looking ahead to future studies of the neutral states, we comment that: 1. In the NMH framework, the HA channel has the third largest cross section. We present the decay patterns of H and A in figure 12 and figure 13 respectively in appendix C. Recall from table 1 that A → hZ is relevant for λ 4 determination, we find that the pp → HA → hh hZ → γγbbbb + − channel only has O(100) events at the future collider with √ s = 100 TeV and L = 30 ab −1 even without considering the backgrounds. Again, the event number can be improved by studying different final states or different decays chain including vertices in table 1.
2. For λ 4 determination, the H ± → hW ± channel has a larger branching ratio for λ 45 < 0. In comparison, H → ZZ has a larger branching ratio λ 45 > 0, which makes the vacuum stable to a higher scale compared with the benchmark point we use in this work. On the other hand, H → W + W − /A → hZ channel dominates for both positive and negative λ 45 as can be seen from the right panel of figure 12 and figure 13. Therefore, theoretically, the HA channel also provides a way to for model discovery and λ 4,5 determination at the 100 TeV collider. Figure 12. Decay region plots for H with BR ≥ 40%. Black region for the di-Higgs channel, blue region for the di-W boson channel and purple region for the di-neutrino channel. From the left panel, di-neutrino/di-h channel dominates at small/large v ∆ respectively, and W -pair channel dominates at the large v ∆ and small m ∆ region. While from the right panel, we observe that di-h (di-Z boson) channel dominates for negative (positive) λ 45 . Figure 13. Decay region plots for A with BR ≥ 40%. Black region for the hZ channel, purple region for the t quark pair channel and blue region for the di-neutrino region.

B Decay rates of h → γγ
Here we briefly review the computation of the ratio R hγγ . The current combined value from ATLAS and CMS at √ s = 7 TeV and √ s = 8 TeV is R hγγ = 1.16 +0.20 −0.18 [65,166], and in the CTHM, the singly-and doubly-charged Higgs particles will contribute to Γ BSM through loop effects. There has been many literatures studying this contribution [23,60,65,71,165], and it was also shown that in the CTHM ( [71,165]), the doubly-charged Higgs particle will give a sizable contribution to the decay rate of h → γγ especially for negative λ 4 and λ 45 due to a constructive interference [23]. Since we choose negative λ 4 and λ 45 in our cases, contributions to the rate from the CTHM can in turn be used to constrain the parameter space of the CTHM. To study this effect, we rewrite the result in refs. [65,192] by using our notation as follows: with α the fine structure constant, g the U(1) coupling, N f c the color factor (N f c =3 for quarks and 1 for leptons), Q f the fermion electric charge, G F the Fermi constant and (i = f, W, H ± , H ±± ). g hW ± W ∓ , g hH ± H ∓ and g hH ±± H ∓∓ , 16 are the couplings given in appendix D. And the loop functions A i are defined as: for τ x < 1. (B.5)

C H and A decays
In this section, we give the dominant decay channels of the neutral Higgs bosons. Note that H → ZZ/W W and A → hZ are relevant for λ 4,5 determination, we see that A → hZ and H → W W can be used for both positive and negative λ 45 , while H → ZZ only works for positive λ 45 . This scenario is different for the fourth channel related to the determination of λ 45 , i.e., H ± → hW ± , which works only for negative λ 45 .