Higgs production at future e+e− colliders in the Georgi-Machacek model

We study how the dominant single and double SM-like Higgs (h) production at future e+e− colliders is modified in the Georgi-Machacek (GM) model. On imposing theoretical, indirect and direct constraints, significant deviations of h-couplings from their SM values are still possible; for instance, the Higgs-gauge coupling can be corrected by a factor κhV V ∈ [0.93, 1.15] in the allowed parameter space. For the Higgs-strahlung e+e− → hZ and vector boson fusion processes e+e−→hνν¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {e}^{+}{e}^{-}\to h\nu \overline{\nu} $$\end{document}, he+e−, the cross section could increase by 32% or decrease by 13%. In the case of associated production with a top quark pair e+e−→htt¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {e}^{+}{e}^{-}\to ht\overline{t} $$\end{document}, the cross section can be enhanced up to several times when the custodial triplet scalar H30 is resonantly produced. In the meanwhile, the double Higgs production e+e−→hhZhhνν¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {e}^{+}{e}^{-}\to hhZ\left(hh\nu \overline{\nu}\right) $$\end{document} can be maximally enhanced by one order of magnitude at the resonant H1,30 production. We also include exclusion limits expected from future LHC runs at higher energy and luminosity and discuss their further constraints on the relevant model parameters. We find that the GM model can result in likely measurable deviations of Higgs production from the SM at future e+e− colliders.

To unravel the symmetry breaking sector it is necessary to measure the interactions of the Higgs boson with itself and other related particles. While this is believed to be JHEP02(2018)007 very challenging at the LHC [27], future lepton colliders provide an avenue to study these interactions at a reasonably good precision due to cleaner environment [28]. There have been several proposals for next-generation e + e − colliders that are under active studies, including the Circular Electron-Positron Collider (CEPC) [29,30], the Future Circular Collider (FCC-ee) [31], the Compact Linear Collider (CLIC) [32,33], and the International Linear Collider (ILC) [34][35][36]. These colliders are planned to operate at a center of mass (CM) energy ranging from about 250 GeV to 3 TeV, thereby making accessible most of dominant production processes of the Higgs boson.
At an e + e − collider of high enough energy one could study simultaneously the interactions of the Higgs boson h with itself and with gauge bosons W ± , Z or even fermions such as the top quark t. This would be very helpful for us to build an overall picture on the symmetry breaking sector and gain a hint on possible physics that goes beyond the SM. The hZZ coupling can be measured via the Higgs-strahlung process e + e − → hZ and the ZZ fusion process e + e − → (ZZ → h)e + e − , while the hW W coupling can be probed via the W W fusion process e + e − → (W W → h)νν. The top Yukawa coupling can possibly be extracted from the associated production process e + e − → htt. And finally, the Higgs pair production processes e + e − → hhνν, hhZ provide an access to the trilinear coupling of the Higgs boson. These processes are generally modified by new interactions or new heavy particles, and precise measurements on them could help us identify the imprints of physics beyond the SM [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53].
In this work we will study the above mentioned dominant single and double Higgs production at future e + e − colliders in the Georgi-Machacek (GM) model [54,55]. The model is interesting because it introduces weak isospin-triplet scalars in a manner that preserves the custodial SU(2) V symmetry. While this symmetry guarantees that the ρ parameter is naturally unity at the tree level, the arrangement of the triplet scalars allows them to develop a vacuum expectation value (VEV) as large as a few tens of GeV. After spontaneous symmetry breaking there remain ten physical scalars that can be approximately classified into irreducible representations of SU(2) V , one quintuplet, one triplet and two singlets. These multiplets couple to gauge bosons and fermions differently. In particular, the couplings of the SM-like Higgs boson h can be significantly modified, and processes involving h receive additional contributions from new scalars as intermediate states. It is a distinct feature of the GM model compared to a scalar sector with only singlet and doublet scalars that the h couplings to fermions and gauge bosons may be enhanced, or always enhanced in the case of the quartic couplings to a gauge boson pair.
The GM model has been extended by embedding it in more elaborate theoretical scenarios such as little Higgs [56,57] and supersymmetric models [58,59], by generalizing it to larger SU(2) multiplets [60] or including dark matter [61,62]. The phenomenology of exotic scalars has previously been studied, including searches for exotic scalars and the application of a variety of constraints on the model parameter space ; in particular, previous works on e + e − colliders [75,80] mainly concentrated on the custodial quintuplet particles. When these exotic scalars are heavy, it is difficult to produce them directly even at LHC, but we will show that they could be probed indirectly at e + e − colliders via modifications to the SM-like Higgs production processes. If the new scalars are light JHEP02(2018)007 enough, they could contribute as resonances in those processes and thus affect them more significantly. In either case, high energy e + e − colliders could provide a viable way to test the GM model. This paper is organized as follows. We recall in the next section the basic features in the Higgs sector of the Georgi-Machacek model. We discuss in section 3 various constraints on the model parameter space coming from current and future LHC runs as well as from low energy precision measurements and theoretical considerations. Then we investigate various SM-like Higgs production processes in section 4 at the 500 GeV and 1 TeV ILC. We reserve for our future work a comparative study of electron colliders operating at various energies and luminosities which requires a detailed simulation of the relevant processes. We finally summarize our main results in section 5. Some lengthy coefficients and Higgs trilinear couplings are delegated to appendices A, B, and C.

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The spontaneous symmetry breaking is triggered by the VEVs Φ = 1 2 v φ / √ 2 and ∆ = 1 3 v ∆ . As usual, the weak gauge bosons obtain masses from the kinetic terms of the scalars with g 2,1 being the gauge couplings of SU(2) L × U(1) Y . Their squared masses are m 2 which should be identified with 1/( √ 2G F ) where G F is the Fermi constant. The parameter ρ = 1 is thus established at the tree level. Since the custodial symmetry is explicitly broken by hypercharge and Yukawa couplings, divergent radiative corrections to ρ will generally appear at one loop within the framework of the GM model [88].
Restricting our discussions to the tree level, the custodial symmetry is respected by the scalar spectra so that the scalars can be classified into irreducible representations of SU(2) V . Excluding the would-be Nambu-Goldstone bosons to be eaten up by the weak gauge bosons, they are decomposed into a quintuplet, a triplet and two singlets. Denoting the real and imaginary parts of the neutral components of the original fields after extracting out the VEVs, the quintuplet states are [86] H ++ 5 = χ ++ , We denote the quintuplet and triplet masses as m 5 and m 3 respectively. At the Lagrangian level, the quintuplet scalars couple to the electroweak gauge bosons but not to fermions (i.e., with H 5 V V but without H 5 ff couplings), while the opposite is true for the triplet scalars (with H 3 ff but without H 3 V V couplings). The two custodial singlets mix by an angle α into the mass eigenstates with c α = cos α and s α = sin α. We assume that the lighter state h is the observed 125 GeV scalar [1,2,89]. The angle is determined by [86] sin 2α = 2M 2 12 where m h,1 are the masses of h and H 0 1 respectively, and in terms of the scalar couplings and VEVs, (2.14)

Constraints on parameter space
There are generally many free parameters in the scalar potential with an extended scalar sector. In this section, we will perform a combined analysis based on theoretical considerations and indirect and direct constraints to obtain the allowed regions for the parameters that are most relevant for our later Higgs production processes. The theoretical constrains are mainly derived from the requirement of perturbativity and vacuum stability [86,87], while the indirect ones cover the oblique parameters (S, T, U ), the Z-pole observables (R b ), and the B-meson observables [86,90]. Among the B-meson observables (B 0 s −B 0 s mixing, B 0 s → µ + µ − , and b → sγ), the decay b → sγ currently sets the strongest bound. All of these constraints have been implemented in the calculator GMCALC [91] for the GM model, which will be applied as our starting point. On top of this we will impose up-to-date direct experimental constraints which cover the searches for a heavy neutral Higgs boson (H 0 1 ), custodial triplet bosons (H 3 ) and quintuplet bosons (H 5 ), and the signal strength analysis of the SM-like Higgs boson (h). For the signal strength analysis, we will also include the constraints expected from future runs of LHC and the proposed ILC. In figure 1 we show how survived points in the α − v ∆ plane evolve with the inclusion of various constraints. With theoretical and indirect constraints alone, a v ∆ as large as 60 GeV is still allowed. When the constraints from direct searches for H 0 1 (black points), JHEP02(2018)007 H 5 (red), H 3 (yellow) and from the Higgs signal strength (green) are included, more and more points are excluded. At this stage, we have v ∆ 40 GeV, and α < 0 is preferred. Also shown in the figure are the future prospects of constraints derived from Higgs signal strength measurements at 14 TeV LHC with 300 fb −1 (green curve), High-Luminosity LHC (HL-LHC) with 3000 fb −1 (magenta) [92], ILC with 1150 fb −1 at 250 GeV (orange) and 1600 fb −1 at 500 GeV (black) [35]. It is clear that a wide parameter space will be within the reach of future ILC operations. In the following subsections we will describe these direct experimental constraints in more detail.

Singlet searches
In the GM model, the heavy neutral Higgs boson H 0 1 can decay to a pair of vector bosons when it is above the threshold. Searches for heavy resonances decaying to a W W [93] and ZZ [94] pair are performed by the ATLAS Collaboration using the data collected at √ s = 13 TeV with an integrated luminosity of 13.2 fb −1 . For the gluon fusion (ggF) and vector boson fusion (VBF) production of the heavy Higgs, the corresponding cross sections are calculated as with gs denoting the couplings in the SM and the GM model. The theoretical cross sections of a SM-like heavy Higgs σ The (gg → H 0 1 ) and σ The (qq → H 0 1 ) have been tabulated in ref. [95], while the branching ratio BR GM (H 0 1 → V V ) is obtained using GMCALC. When m 1 > 2m h , H 0 1 can also decay into a Higgs pair hh, which may greatly enhance the Higgs pair production at the LHC. The cross section for resonant production of a Higgs boson pair is given by [96] where BR GM (H 0 1 → hh) is also calculated by GMCALC. Recently, a search for resonant Higgs boson pair production (H 0 1 → hh) has been performed by the CMS Collaboration [97] in the bb νν final state.
In figure 2 we show in the v ∆ −m 1 and α−m 1 planes the survived points upon applying the constraints from the direct searches H 0 1 → W W , H 0 1 → ZZ, and H 0 1 → hh. While the H 0 1 → W W, ZZ searches remove only a small portion of points, a considerable portion is excluded by the H 0 1 → hh search. Due to large variations of BR(H 0 1 → hh) in our scan [96], no clear dependence of the exclusion bounds on v ∆ and α is visible.

Triplet searches
The signature of a neutral triplet Higgs boson H 0 3 has been considered in ref. [98]. Without direct couplings to gauge bosons, the promising signature is gg [98]. The charged triplet Higgs boson H + 3 can decay into τ + ν for m 3 < m t or into tb for m 3 > m t + m b [67]. We therefore consider the following direct searches: • Search for a CP-odd Higgs boson H 0 3 decaying to hZ [99,100].
• Search for a heavy Higgs boson H 0 3 decaying to a top quark pair [101]. • Search for charged Higgs bosons

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Our results are shown in figure 3 in the v ∆ − m 3 and α − m 3 planes. Among the constraints from those searches, H 0 3 → hZ sets the most stringent one. In particular, in the mass region 200 m 3 500 GeV, where BR(H 0 3 → hZ) is dominant or relatively large, v ∆ can be pushed down as low as 10 GeV under certain circumstances.

Quintuplet searches
Being independent of the singlet mixing angle α, the constraints on the quintuplet scalars are only sensitive to the VEV v ∆ and their mass m 5 . In ref. [98] the constraint on v ∆ has been obtained as a function of the exotic Higgs boson mass via various channels for the additional neutral scalars in the GM model. In this paper, we adopt the constraints from the decay channels H 0 5 → γγ and H 0 5 → ZZ through the VBF mechanism. In refs. [104,105]  mass were studied in the GM model in ref. [106] by recasting the ATLAS measurement of the cross section for the like-sign diboson process pp → W ± W ± jj. The W + W + W − W − vertex is effectively modified by mediations of the doubly-charged Higgs bosons H ±± . That the relevant W ± W ± H ∓∓ vertex is proportional to v ∆ can be used to exclude parameter space on the plane of v ∆ and m 5 . In this work we also take into account the latest search for like-sign W boson pairs by the CMS [107]. Additional subsidiary constraints are as follows:

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• An absolute lower bound on the doubly-charged Higgs mass from the ATLAS like-sign dimuon data was obtained in ref. [108], which gives m 5 76 GeV.
• An upper bound on s H for m 5 ∼ 76 − 100 GeV can be obtained using the results of a decay-model-independent search for new scalars produced in association with a Z boson from the OPAL detector at the LEP collider [109].
In figure 4, we summarize the constraints from searches for quintuplet Higgs bosons on the v ∆ − m 5 plane. In the low mass region 76 GeV < m 5 < 110 GeV, the most stringent constraint comes from the neutral Higgs boson decay H 0 5 → γγ through the VBF production. In the mass interval 110 − 200 GeV, the like-sign diboson process pp → W ± W ± jj via the doubly-charged Higgs boson H ±± 5 gives the best bound. Above 200 GeV, the most severe constraint is set by the latest search for like-sign W boson pairs, which could exclude v ∆ 15 GeV when m 5 ∼ 200 GeV.

Higgs signal strengths
The signal strengths of the SM-like Higgs boson production and decay in various channels can provide significant constraints on its couplings to the SM particles in the GM model [111]. The signal strength for a specific production and decay channel i → h → f is defined as where σ i (σ SM i ) is the reference value (SM prediction) of the Higgs production cross section for i → h, and BR f (BR SM f ) the branching ratio for the decay h → f . We include JHEP02(2018)007 the production channels via the gluon fusion (ggF ), the vector boson fusion (VBF), the associated production with a vector boson (V h) and with a pair of top quarks (tth), and the decay channels h → γγ, ZZ, W ± W ∓ , τ ± τ ∓ , bb. With the experimental values of µ exp X and standard deviation ∆µ exp X [112], we build a χ 2 value for each allowed point as where the sum extends over all channels mentioned above. From eqs. (4.1), (4.6), we are aware that the SM-like Higgs couplings involved in the signal strengths depend only on the triplet VEV (v ∆ ) and the singlet mixing angle (α). Therefore the constraints on v ∆ and α can be directly extracted from data without specifying other parameters. In figure 5 we show the scatter plot of χ 2 values on the α − v ∆ plane within a 2σ range. It is clear that the measurement of the Higgs signal strengths is most sensitive to the region with large v ∆ and large |α|, where large deviations of Higgs couplings from the SM take place. Hence the large χ 2 region, e.g., v ∆ ∼ 50 GeV and α ∼ −40 • would be excluded by future operations of LHC [76].

Future experimental constraints
For completeness we also presented in figure 1 the constraints of the Higgs signal strength χ 2 fit on the α − v ∆ plane based on the projection results from 14 TeV LHC with an integrated luminosity of 300 fb −1 (LHC@300) and 3000 fb −1 (HL-LHC@3000) [92] and from ILC with an integrated luminosity of 1150 fb −1 at 250 GeV (ILC250) and 1600 fb −1 at 500 GeV (ILC500) [35]. The LHC (HL-LHC) result is performed on the ATLAS detector with only statistical and experimental systematic uncertainties taken into account. The expected precision is given as the relative uncertainty in the signal strength with the central values all assumed to be unity. In principle, this assumption applies only to the SM, but we employ these anticipated results as a reference so that we could be clear to what extent they can impose a constraint on the parameter space. It is clear from figure 1 that the constrains from LHC@300, HL-LHC@300, ILC250, and ILC500 gradually become more and JHEP02(2018)007 more stringent. Basically speaking, the deviations in Higgs couplings are determined by α and v ∆ . Our fitting results show that LHC@300 (HL-LHC@3000) could approximately exclude v ∆ 30 (20) GeV while ILC250 and ILC500 could further push it down to about v ∆ 10 GeV.
Let us consider the scale factors κ hV V and κ hff as an example under the assumption that no obvious deviations in Higgs couplings from SM values will be observed. At LHC@300, κ hV V and κ hff could be constrained within the ranges [0. 92 On the other hand, if the on-going LHC observes a certain hint of Higgs coupling deviation , for instance, κ hV V > 1 for the most optimistic case, the future operation of ILC will be hopeful to confirm it. In this way, the GM model would be strongly favored by the deviation κ hV V > 1. 4 Higgs production at e + e − colliders In this section we will study the dominant single and double SM-like Higgs production at future e + e − colliders in the GM model. For comparison we first reproduce in figure 6 the dominant Higgs production cross sections in the SM. The Higgs-strahlung (HS) process (e + e − → hZ) is dominant for a CM energy √ s < 500 GeV. At higher energy, the Higgs production is dominated by the W W fusion process (e + e − → hν eνe ), while the ZZ fusion process (e + e − → he + e − ) also becomes significant. The subdominant processes such as e + e − → htt, e + e − → hhZ and e + e − → hhν eνe provide access to the top Yukawa coupling and the Higgs trilinear self-coupling. It is clear that due to the s-channel topology (see figures 7, 10, 14), the hZ, htt, and hhZ production cross sections become maximal near the thresholds and decrease as collision energy goes up. On the contrary, the VBF (hν eνe , he + e − , and hhν eνe ) cross sections increase as ln √ s due to their t-channel topology (see figures 7, 16). These dominant processes can be divided into three types according to the couplings involved, namely the Higgs couplings to gauge bosons (figure 7) and to the top quark (figure 10), and the trilinear Higgs self-couplings (figures 14, 16), which we now investigate in detail for the GM model.

Production via Higgs-strahlung and vector boson fusion
In figure 7 we depict the Feynman diagrams for single Higgs production at e + e − colliders that involves only Higgs couplings to weak gauge bosons. The amplitudes for both HS and VBF processes are modified in the GM model by the same ratio of the Higgs-gauge couplings from the SM,   where c H and s H are defined in eq. (2.11) in terms of v ∆ . We can thus extract κ 2 hV V by measuring these cross sections and set constraints on the parameters v ∆ and α, The predicted value of κ hV V in the v ∆ − α plane is shown in figure 8. It is obvious that an O(10%) deviation of κ hV V from unity is still viable. Although the allowed κ hV V is in a range of about 0.93-1.15, most of the survived points tend to have κ hV V > 1. The JHEP02(2018)007 LHC@300 will mostly exclude κ hV V 1.1, while HL-LHC@3000 could probe down to κ hV V 1.05. The scale factor κ hV V is expected to be measured at future e + e − colliders with high precision. For example, the measurements for the hV V couplings may reach an accuracy of 1% at CEPC (250 GeV, 5 ab −1 ) and ILC (500 GeV, 500 fb −1 ) [30,35]. Hence, the GM model could be probed indirectly if a large enough deviation of κ hV V from unity is measured. Especially, a measured κ hV V > 1 would be strong evidence in favor of the GM model. If κ hV V turns out to be consistent with unity, a precise measurement of it would put a stringent bound on the GM model parameter space.
The cross section for the HS process is [113] σ(e + e − → Zh) = where the Z couplings to the fermion f of electric charge 3 , with I f 3 = ±1/2 being the third weak isospin of the left-handed fermion f , and s 2 W = sin 2 θ W with θ W being the Weinberg angle. And β = (1 − r Z − r h ) 2 − 4r Z r h is the usual two-body phase space function with r h,Z,W ≡ m 2 h,Z,W /s, etc. The total cross section for the W W (ZZ) fusion process is [114] for the W W (ZZ) fusion respectively, and relative corrections compared to the SM δσ/σ SM = κ 2 hV V − 1 are the same and independent of the collision energy. From figure 9, we see that the cross sections could maximally increase by 32% or decrease by 13% with the current constraints. Even if no clear deviation would be found at LHC@300 or HL-LHC@3000, an increase of up to 10% in the HS and VBF cross section would still be possible at the ILC.

Associated production with top quarks
The associated Higgs production with a top quark pair is an important process to measure the top quark Yukawa coupling at a linear collider [115][116][117]. Compared to the SM case, the existing interactions are modified and in addition there is a new contribution in the GM model that is mediated by the CP-odd heavy Higgs H 0 3 , see figure 10. The scale factor to the SM hf f coupling is, κ hff = c α /c H , and those involving the custodial triplet H 3 are which appear in the Feynman rules as follows: where p h (p 3 ) is the incoming momentum of h (H 0,+ 3 ). The scanned results for these κs are shown in figure 11 in the v ∆ − α plane. For the current constraints, we see that a deviation from unity as large as O(±10%) is still allowed for κ hff . The scale factor κ H 0 3 ff does not depend on α, and its magnitude can maximally reach about 0.54 and vanishes in the limit of v ∆ → 0, while κ H 0 3 hZ lies in the interval of −0.2 to 0.6. The future operation of LHC@300 will be capable of excluding points with κ hff 0.975 and |κ H 0 3 ff | 0.35, while ILC500 has the ability to constrain κ hff ≈ 1, |κ H  The cross section in the GM model can be adapted from those in the SM and MSSM [116,117] dσ(e + e − → htt) (4.8) Here σ 0 = 4πα 2 QED /(3s), α QED is the fine structure constant, N c = 3, E h the Higgs boson energy. The explicit expressions for the coefficients G i (i = 1, · · · 7) are given in appendix A. In figure 12 we present the cross section for e + e − → htt as a function of the H 0 3 mass m 3 at √ s = 500, 1000 GeV. In the parameter region 2m t < m 3 < √ s − m h , the subprocess e + e − → hH 0 3 , H 0 3 → tt is resonant, where the cross section can be strongly enhanced by several times. When the mass of H 0 3 is far above the resonant region, the cross section reduces gradually to the SM value. Interestingly, in the resonant region with enhanced cross section, we actually have κ hff 1 for most of the survived points, while above this region the cross section decreases with the decrease of κ hff . Combined with JHEP02(2018)007 Figure 13. Same as figure 11, but for κ hhh , κ hhV V , κ H 0 1 hh and κ H 0 1 V V figure 11, most of points with κ hff < 1 will be excluded by LHC@300 and HL-LHC@3000. Hence, large decrease of the htt production might not be possible. On the other hand, for H 0 3 resonantly produced, most of allowed points actually have κ hff ≈ 1. Therefore, even if no large deviation in the Higgs coupling is observed at ILC250 or ILC500, significant enhancement of the htt production may still be viable. At CLIC the expected accuracy for cross section is about 8.4% with an integrated luminosity of 1.5 ab −1 at √ s = 1.4 TeV [33], and at ILC the accuracy could reach 28% (6.3%) at 500 (1000) GeV [35]. There is thus a good chance to test the htt production at these high energy machines.

Double Higgs production
To study the Higgs self-interactions in the symmetry breaking sector, it is indispensable to measure the Higgs pair production at future e + e − colliders. In this subsection we consider possible production mechanisms in the GM model. These processes include the double Higgs-strahlung process e + e − → hhZ in figure 14 and the Higgs pair production via the W boson fusion e + e − → hhν eνe in figure 16 while ignoring the much smaller Z boson fusion [33,34]. These processes involve the scale factors

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κ H 0 1 V V shown in eq. (3.2), and κ hhh = g hhh /g SM hhh , κ H 0 1 hh = g H 0 1 hh /g SM hhh , where g SM hhh = 3m 2 h /v, and g hhh , g H 0 1 hh are given in appendix B. In figure 13, we show the scanned results for these scale factors in the allowed parameter space. Compared to their SM counterparts, the hhV V coupling is always enhanced due to κ hhV V ≥ 1, whereas the hhh coupling can even change its sign. For those points with κ hhh < 0, LHC@300 will exclude most of them while HL-LHC@3000 could eliminate them altogether. The precision expected to be reachable at the ILC for the measurement of g hhh is of O(10%) [36], so the effect of the GM model should be observable. For the new couplings involving H 0 1 , the magnitude of κ H 0 1 hh can be so large that the processes are significantly enhanced but perturbation theory may not apply there, while the scale factor κ H 0 1 V V ranges from −0.2 to 0.6. Considering the limit expected to be available at LHC@300 (HL-LHC@3000), κ H 0 1 hh 10 (κ H 0 1 hh 5) should be satisfied.

The function A can be expressed in the form [119]
where a = 1 2 Herer i (i = h, Z, H 1 , H 3 ) includes the total decay width of the particle i in its mass squared, i.e.,r i = (m 2 i − im i Γ i )/s when a resonance is crossed over. The coefficients f and g in eq. (4.11) are  The double Higgs boson production is sensitive to the triple Higgs coupling g hhh , which cannot be probed by the single Higgs boson production. In addition, the heavy CP-even Higgs boson H 0 1 contributes, thereby enabling sensitivity to the H 0 1 hh coupling as well. The total cross section is shown in figure 15 at the energy √ s = 500 GeV and 1 TeV. Compared to the SM case, the on-shell production of heavy scalars H 0 1 and H 0 3 followed by decays H 0 1 → hh and H 0 3 → hZ plays a dominant role in the resonant region of the parameter space, where the cross section can increase by more than one order of magnitude. Since in the resonance region, most of the currently allowed points have κ hhh ≈ 1, the future measurements at LHC@300 and HL-LHC@3000 would be hard to exclude such points. In the non-resonance region, the cross section at √ s = 500 GeV is positively correlated with  Figure 16. Feynman diagrams for double Higgs production from W W fusion at e + e − colliders. κ hhh , leading to enhanced cross section with κ hhh > 1 or suppressed cross section with κ hhh < 1. The cross section at √ s = 1 TeV can be either enhanced or suppressed even with κ hhh ≈ 1.

Double Higgs from vector boson fusion
Besides the resonant W W fusion for single h production, there exists non-resonant W W fusion production of a pair of h, e + e − → hhν eνe , shown in figure 16. The GM model modifies existing interactions in the SM and introduces new contributions due to H 0 1 and H ± 3 exchanges. The cross section in the effective W approximation can be written as [119,120] σ where x min = 4r h (= 4m 2 h /s), the differential luminosity function is [119] dL 16) and the cross section for the subprocess is [119] where the functions F i (i = 1, . . . , 5) are reproduced in appendix C and β h = (1 − 4r h ) 1/2 . Note that here r i (i = h, Z, H 1 , H 3 ) are defined with respect toŝ = xs, e.g., r 1 = m 2 1 /ŝ. The total cross section for e + e − → hhν eνe is shown in figure 17 at √ s = 500 GeV and 1 TeV respectively. It varies in a wide range as κ hhh and κ H 0 1 hh do, indicating its sensitivity to the values of trilinear couplings. Similarly to the case of the double Higgs-strahlung process, the cross section is strongly enhanced in the parameter range where the fusion subprocess, W W → H 0 1 → hh, is resonant also with κ hhh ≈ 1. Out of the resonant region the cross section decreases with increasing κ hhh due to destructive interference between the scalar and gauge parts of the amplitude. From figure 13, one sees that κ hhh < 1 would not be favored by HL-HLC@3000. In this case, we expect a smaller cross section in the non-resonance region.

Conclusions
In this paper, we have investigated the dominant single and double SM-like Higgs (h) production at future e + e − colliders in the Georgi-Machacek (GM) model. We comprehensively considered various constraints that are currently available, including theoretical, indirect, and direct experimental constraints on the model parameters. In particular, we updated the constraints from searches for heavy singlet (H 0 1 ), triplet (H 3 ), and quintuplet (H 5 ) Higgs scalars and from SM-like Higgs signal strengths with the latest LHC results. The GM model parameter space now becomes more tightly constrained, for instance, only v ∆ 40 GeV with −30 • α 0 • is allowed now.
Our analysis of the Higgs production has been done in the allowed parameter regions established as above. Besides modifications to the SM couplings, the SM-like Higgs production receives new contributions from new scalars such as H 0 1 , H 0 3 , and can deviate significantly from the prediction in the SM. Our numerical results have the following key features: • For single Higgs production such as e + e − → hZ, hν eνe , he + e − , the production mechanisms are the same as in SM with the only modification occurring in the Higgsgauge boson coupling hV V . Their cross sections are modified by a factor ranging from 0.86 to 1.32.
• For associated Higgs production with a top pair e + e − → htt, there is a new contribution via e + e − → hH 0 becomes resonantly produced, thus enhancing the cross section by up to three times.
• For the double Higgs-strahlung process e + e − → hhZ, both H 0 1 and H 0 3 contribute. When resonantly produced, they can lead to an order of magnitude enhancement in the cross section. In the non-resonant region, large deviations are still possible due to a large modification to the trilinear Higgs coupling.

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• For the double Higgs production via vector boson fusion e + e − → hhν eνe , both H 0 1 and H ± 3 enter but only H 0 1 can be on-shell produced. In the resonant region more than an order of magnitude enhancement is viable, and in the non-resonant region the cross section can maximally increase or decrease by an order of magnitude.
Anticipating that LHC will be operated at slightly higher energy and with larger integrated luminosity, we have also estimated its future impact on Higgs couplings and production at electron colliders. We found that LHC@300 and HL-LHC@3000 have the ability to exclude some of the parameter space that is currently allowed. When the new scalars in the GM model are light enough to be resonantly produced at electron colliders, large enhancement in the SM-like Higgs production is possible though its couplings are close to the SM values. In the non-resonance region, the double Higgs production channels are expected to deviate significantly from the SM under the strict future bounds from LHC. If the on-going LHC experiments could observe some deviations of the Higgs couplings, future electron colliders would be capable of confirming it; if not, precise measurements of the Higgs properties at future electron colliders could probe most of the currently allowed parameter space of the GM model. Compared to other popular models such as MSSM and THDM, the GM model can enhance or suppress Higgs couplings to vector bosons and fermions, and drastically change the trilinear Higgs coupling from the SM prediction. The modification of these Higgs couplings leads to distinguished phenomenology for Higgs production processes at e + e − colliders. Furthermore, light scalars such as H 0 1 and H 0 3 can be on-shell produced in certain processes, which provides a good chance to hunt new particles. We thus expect that the model could be tested with high precision and be discriminated from some other new physics scenarios.

JHEP02(2018)007
The other five coefficients all involve the Higgs radiation from the Z boson including its interference with the radiation from the top quark: The relevant h couplings are defined in terms of their κ factors: The kinematical variable β t is

JHEP02(2018)007
C Coefficients for σ(e + e − → hhν eνe ) The coefficients used in the calculation of e + e − → hhν eνe are [119] F 1 = 8[2r W + (r h − r W ) 2 ]l W − 4β h (1 + 2r h − 2r W ), Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.