The depleted Higgs boson: searches for universal coupling suppression, invisible decays, and mixed-in scalars

Two simple ways by which the standard signals of the Standard Model Higgs boson can be depleted are: its couplings to fermions and gauge bosons can be suppressed by a universal factor, and part of its branching fraction can be drained into invisible final states. A large class of theories can impose one or both of these depletion factors, even if mild, by way of additional scalar bosons that are singlets under the Standard Model but mix with the Higgs boson. We perform a comprehensive survey of the present status of the depleted Higgs boson, and discuss future prospects for detecting the presence of either depletion factor. We also survey the constraints status and future detection prospects for the generic case of extra mixed-in scalars which generically lead to these depletion factors for the Higgs boson. We find, for example, that precision study of the Higgs boson in many cases is more powerful than searches for the extra scalar states, given the slate of next-generation experiments that are on the horizon.


I. INTRODUCTION
One of the biggest triumphs of the Standard Model (SM) is the explanation of the electroweak symmetry breaking via a CP-even complex scalar H ∼ (1, 2, 1 2 ) which further predicts the existence of a neutral Higgs boson h SM that is part of the neutral component of the SU (2) L doublet H. So far all the measured properties of the ∼ 125 GeV physical Higgs boson that was discovered in 2012 at the Large Hadron Collider (LHC) are consistent with the SM predictions and therefore the observed Higgs boson h is often identified as the SM Higgs boson h SM .
It is important to note, however, that as the Higgs measurements get more precise there is a well-motivated possibility that the observations can deviate from the SM predictions thereby pointing towards the presence of a more complicated Higgs sector. In this paper we study the prospect of the observed Higgs boson h having a universal depletion factor δ, suppressing all its couplings to the SM states, along with an invisible width. † The invisible width constitutes a second depletion factor for the Higgs boson because its standard decay final state branching fractions are depleted due to the additional invisible mode.
To set notation, the SM Higgs boson h SM is related to the observed Higgs boson h in the following way: where · · · include contributions from additional physical Higgs states that can arise from extended Higgs sectors. In other words, we identify the label "Higgs boson" h with the h125 discovered resonance and not necessarily with the Platonic ideal Standard Model Higgs boson h SM . Below, we obtain constraints on the universal depletion factor δ, dependent on the invisible width of h, from various searches for the ∼125 GeV Higgs boson at the LHC while remaining agnostic to the particulars of the extended Higgs sectors. We also later reinterpret some projections at the International Linear Collider (ILC) and the High Luminosity LHC (HL-LHC) from the invisible decays of h. We emphasize that the relation in eq. (1.1) can naturally arise from many extended Higgs sectors. As a concrete example for an extended Higgs sector that leads to a universal depletion factor δ and an invisible width for h, we first consider an extension to the SM with a real scalar S ∼ (1, 1, 0) that mixes with the SM Higgs boson h SM to give rise to two physical mass eigenstates: the observed Higgs h and an exotic Higgs φ. We take the singlet scalar S to only have invisible decays which translates to an invisible width for h via its mixing with S. SM extensions with a singlet scalar were extensively studied in the literature, see e.g. refs.  and the references therein.
For illustration purposes, we only restrict ourselves to the mass of the exotic Higgs φ to be in between m h /2 and 2m h such that the decays h → φφ and φ → hh are kinematically forbidden. As we will see below that after re-examining the parameter space with the latest bounds, the indirect constraints from the precision probes for the observed Higgs boson h at the LHC are the dominant ones in general compared to the bounds from the searches † See also refs. [1,2] for related studies on the impact of two universal depletion factors on Higgs signal strength measurements.
for additional neutral Higgs states, and from the oblique parameters S, T, and U . This is in accord with the findings in refs. [16,17] for m h /2 ≤ m φ ≤ 2m h specifically for the case where h does not have any invisible decays.
We also show some future projections at the ILC. As a further extension to this case, we also briefly consider the case with N such scalars that are assumed to mix equally among themselves, and with the SM Higgs boson. Once again, as we will see below, we find the indirect bounds from precision probes for the observed Higgs boson h moderately constrain the parameter space, which gets stronger as N and/or the invisible width of h gets larger.
The remaining sections of this paper are structured as follows. In Section II, we define "the depleted Higgs boson" -Higgs boson with a universal suppression and invisible width -and obtain its production cross-sections, total width, and branching ratios relative to that of the 125 GeV Higgs in the SM. In Section III we study the constraints on the universal depletion factor δ, that varies with the invisible width of h, from the latest LHC precision probes for the ∼ 125 GeV Higgs boson in various decay channels. We also consider some example future projections from ILC and HL-LHC. In Section IV, we consider the SM extension with a real singlet scalar, and in Section V we reexamine the present direct and indirect bounds along with some future projections from collider searches, and the current constraints from the Peskin-Takeuchi S, T, and U parameters. We also briefly consider a Higgs sector with N real singlet scalars, under some simplifying assumptions, and study the implications of direct and indirect searches on such a scenario in Section VI. Finally, we end with some concluding remarks in Section VII.

WIDTH
Consider the Higgs boson h with its mass of ∼125 GeV and with a universal depletion factor δ for all its couplings to the SM states, such that the production cross-sections of h are given by where σ 125 SM are the production cross-sections for 125 GeV Higgs in the SM. Additionally, assume that the 125 GeV Higgs also has an invisible width that is parameterized as δ 2 Γ inv . Therefore, the total width of h is where Γ 125 SM stands for the total width of the 125 GeV Higgs boson in the SM. Upon defining eq. (2.2) can be reexpressed as Given these definitions, the ratio of the branching ratio B h j of h → j th SM final state to that of the corresponding branching ratio in the SM B 125 SM,j , and the invisible branching ratio B h inv can be entirely expressed in terms of just two parameters − δ and κ inv : (2.6) Note that in the limit δ → 0 the SM is completely recovered (independent of κ inv or Γ inv ). Although, at this point, the appearance of the terms δ and Γ inv (or equivalently κ inv ) might seem ad-hoc, we later consider the case where the SM is extended by a real scalar S ∼ (1, 1, 0) that decays invisibly and mixes with the SM Higgs boson h SM . In which case, δ will be the sine of the mixing angle between h SM and S, and Γ inv is the invisible decay width of S. More generally, the terms δ and κ inv naturally appear in many models beyond the SM, since these "singlets" can represent any number of possible states with exotic charges in a sector beyond the SM. Furthermore, we will see that the constraints on δ and κ inv that come just from the precision probes for the 125 GeV SM Higgs boson moderately constrain the extensions to the Higgs sector. The case with N ≥ 1 real singlet Higgs boson(s) that we later consider will illustrate that.

III. CONSTRAINTS ON DEPLETION FACTORS δ AND κ inv
We now turn to constraints on the 2-dimensional parameter space of (δ, κ inv ) that come from searches by ATLAS and CMS collaborations at the LHC for the observed Higgs boson decaying invisibly or to the SM final states. From invisible searches, a bound is usually reported on the invisible branching ratio B inv of the Higgs boson multiplied with its production cross-section relative to the production cross-section in the SM σ/σ SM . In particular, an upper bound is reported on µ inv † which we define as 1) † For convenience, we define the invisible branching ratio of the Higgs boson multiplied with its production cross-section relative to the production cross-section in the SM as µ inv , which is not to be confused with the signal strength modifier that is defined in eq. (3.4).  [25] which in our case can be written as (see eq. (2.1)) Therefore, using eq. (2.6), the above equation imposes the following constraint on (δ, κ inv ) parameter space Table 3.1 shows some recent LHC upper bounds on the quantity µ inv at 95% CL by ATLAS and CMS experiments. As for visible final states, precision probes for the decay of the 125 GeV Higgs boson to a j th SM final state often report the measured signal strength modifier µ defined as where +σ + µ , −σ − µ account for various systematic and statistical uncertainties on µ. In our case, the above equation also provides a constraint on (δ, κ inv ) parameter space (see eqs. (2.1) and (2.5)), which we obtain by solving for δ as a function a κ inv (or vice versa) from (3.5) Here, we require χ 2 = 5.99 which corresponds to a fit with 95% confidence level with 2 degrees of freedom [26], and we simply take σ µ = (σ + µ + σ − µ )/2 (as we will be concerned here with the cases where the asymmetry, , is either small or absent).  1.2 ± 0.6 [38] for the decays of the 125 GeV Higgs boson to various SM final states. Figure 3.1 shows the most recent bounds from the LHC precision probes for the 125 GeV Higgs boson h in various decay channels at 95% CL. In each decay channel of h, only the strongest bound is shown in the figure. The shaded regions with dashed and solid borders show the bounds obtained from the searches for h to invisible and SM final states (excluding h → γγ), respectively. The bound obtained from the search for h → γγ is shown by a dotted line. Here, a dotted line is used to emphasize that the h → γγ is a loop-induced process, and is therefore more sensitive to new physics contributions.
The future searches, see e.g. refs. [39][40][41][42][43][44][45][46], for h further constrain this parameter space, and as candidate projections, we show the dash-dotted lines corresponding to the 95% CL expected sensitivity for the invisible decays of h at the High Luminosity LHC (HL-LHC) with 3 ab −1 of integrated luminosity [40] and at the International Linear Collider (ILC) with Silicon detector at √ s = 250 GeV with an integrated luminosity of (0.9, 0.9) ab −1 for (e − L e + R , e − R e + L ) respectively and beam polarization of (80, 30)% for (e − , e + ) respectively [46]. Therefore, if the 125 GeV Higgs boson has a universal suppression for all its couplings to SM fermions and gauge bosons along with an invisible width, it is evident from Figure 3.1 that this scenario is moderately constrained from the current LHC searches. And, the future precision studies of the already discovered Higgs boson should be able to further constrain or find evidence for such a scenario with depletion factors (δ, κ inv ). Thus the precision probes for the 125 GeV Higgs boson provide for an indirect, and often the most powerful probe (as we will see below for few examples), for a large class of theories with depletion factors (δ, κ inv ) for the 125 GeV Higgs and additional physical Higgs states.
FIG. 3.1: Current 95% CL bounds on the parameter space of (δ, κ inv ) from ATLAS and CMS precision probes for 125 GeV Higgs boson in various search channels. Out of the bounds obtained from the reported values of µ inv (defined in eq. (3.1)) and the signal strength modifier µ from Tables 3.1 and 3.2, respectively, only the strongest bounds in each search channel are shown here. The shaded regions with dashed and solid borders show the bounds obtained from the searches for h to invisible and SM final states (excluding h → γγ), respectively. The bound from h → γγ is shown by a dotted line to emphasize that the process is loop-induced and therefore more sensitive to new physics contributions. The dash-dotted lines show the projected 95% CL sensitivity for invisible h decays at 250 GeV ILC and high luminosity LHC with nominal running assumptions as discussed in the text.

IV. REAL SINGLET SCALARS EXTENSION
We now consider a model with the SM particle content and an additional real scalar Higgs S that is a singlet under the SM gauge group with the following scalar potential ¶ : where H is the SM Higgs doublet with a weak hypercharge Y = 1/2, and m 2 H is positive such that H acquires a non-zero vacuum expectation value (VEV). The singlet Higgs S is ¶ We treat eq. (4.1) as an effective scalar potential and therefore the tadpole term for S that naively seems to follow from eq. (4.1) does not exist in the full theory (see e.g. refs. [3,4]). assumed to have invisible decays, governed by the interaction Lagrangian: where ψ ∼ (1, 1, 0) is a very light hidden-sector fermion with a Dirac mass m ψ . Therefore, the width for S to decay to the invisible states S → ψψ is Γ inv (m S ) λ 2 ψ m S /8π, with the assumption that m ψ m S . At the minimum of the scalar potential, we assume S = 0, and we use SU (2) L gauge freedom such that only the neutral component H 0 of the SM Higgs doublet acquires a VEV. In unitary gauge, we take . After plugging in for H and expanding around the VEV, eq. (4.1) becomes (4.5) The squared mass matrix can then be diagonalized by a unitary matrix parameterized by a mixing angle which leads to the physical mass eigenstates h, φ that are admixtures of the gauge eigenstates h SM , S: with their corresponding squared masses where m + is the mass of the heavier state and m − is the mass of the lighter state. We take h to be the physical Higgs boson h with mass m h 125 GeV that was discovered at the LHC in 2012, and the other mass eigenstate φ to be the exotic Higgs. From eqs. (4.6) to (4.9), note that as µ S in eq. (4.1) is tuned down to 0, the mixing angle ω vanishes and the mass eigenstates are same as the gauge eigenstates with masses 2m 2 H ( 125 GeV) and m 2 S . For purposes of illustration, in this paper, we only consider the possibility of having m φ in the vicinity of m h , specifically m h /2 < m φ < 2m h , so that the decays h → φφ and φ → hh are kinematically forbidden.
Due to the mixing of the gauge eigenstates h SM and S, the physical states h, φ can now decay to SM states and to invisible states with widths Γ h,φ SM and Γ h,φ inv respectively. Apart from the masses m ± (of which one is already known to be ∼125 GeV), the widths, the production cross-sections and branching fractions of h, φ can be expressed in terms of only two free parameters δ and κ inv that are defined as: In terms of Γ SM (total width for h SM to SM states) and Γ inv (width for S → ψψ): where Γ inv is treated as a free parameter, and Γ 125 SM = Γ SM (125 GeV). Also, in terms of the production cross-section σ SM of h SM , the production cross-sections for h, φ are We can now compare the production cross-sections times branching ratios of h, φ to the ones in the SM. In order to do so, we first note that the total widths Γ h,φ of h, φ are related to the SM width Γ SM via: where we have defined Using these results and definitions, the ratio of branching ratios of h, φ into j th SM final so the production rates of h, φ in j th SM final state is Using the above equations, we can now obtain the current constraints/future sensitivities for real singlet scalar extension to the SM that impose the depletion factors (δ, κ inv ) on the 125 GeV Higgs boson h and also predict an exotic Higgs φ. The indirect constraints and projections from the precision probes for the 125 GeV Higgs boson h were obtained in the previous section (Section III). Whereas, the current constraints are considered in the next section (Section V) along with some future projections for (δ, κ inv ) from the direct searches for the exotic Higgs boson δ with m h /2 ≤ m φ ≤ 2m h (for simplicity and easy compatibility with precision electroweak constraints). And, we will see that the precision probes for the observed Higgs boson h are typically much more constraining than the direct searches for the exotic Higgs φ.

V. CONSTRAINTS ON REAL SINGLET SCALARS
As mentioned above, in the present case the sine-squared of the mixing angle between the gauge eigenstates h SM and S gives rise to the universal depletion factor δ for the couplings of the physical 125 GeV Higgs boson h, and the invisible width Γ inv of S gives rise to the invisible width of h. Therefore, all of the constraints on (δ, κ inv ) from the precision probes for the 125 GeV Higgs boson, considered in Section II, are directly applicable for (δ, κ inv ) in the case of real scalar singlet extension to the SM.
Apart from the indirect constraints that come from the precision probes for the 125 GeV Higgs boson, there are also additional bounds on the exotic Higgs φ for various values of its mass m φ from precision electroweak observables, and also from the collider searches for an additional neutral Higgs boson over a wide range of masses.
First, let us consider the bounds from precision electroweak observables, in particular from the Peskin-Takeuchi parameters S, T , and U [47]. To that end we begin by noting the one-loop contribution of the SM Higgs with mass m to the massive vector boson (V = W, Z) propagators: where α e is the fine structure constant, s W (c W ) is the sine (cosine) of the weak-mixing angle, and A 0 (m 0 ), B 0 (p 2 ; m 1 , m 0 ), and B 00 (p 2 ; m 1 , m 0 ) are the Passarino-Veltman functions following the conventions of ref. [48]. Then the predictions for the Peskin-Takeuchi parameters in the real singlet scalar extension to the SM are given by: In order to finally obtain a bound on δ, we check the compatibility of the model predictions with that of the experimental measurements. In particular, for a chosen m φ , we compute the χ 2 -value, which is a function of δ, using with y i = (S −Ŝ, T −T , U −Û ), and the covariance matrix elements V ii = σ 2 i and V ij = ρ ij σ i σ j (i = j). And, solve for δ after requiring χ 2 = 7.81 corresponding to a fit with 95% confidence level (or a p-value of 0.05) with 3 degrees of freedom [26]. Figure 5.1 shows the bound obtained from Peskin-Takeuchi parameters using the procedure detailed above on δ as a function of m φ . Actually, for a fixed significance level, we find a disagreement in comparing our results with the ones obtained in refs. [16,17]. The reason for this disagreement is due to an additional factor of −1 that is included in the vac- uum polarization function Π V V which can be inferred from (eqs. (24) and (25) of) ref. [15] § which seemed to have propagated into the results in refs. [16,17]. Taking Π V V → −Π V V in eq. (5.1), we exactly reproduce the bounds from S, T, and U parameters reported in refs. [16,17]. To check the correctness of the sign in eq. (5.1), we have computed the oneloop contribution of the SM Higgs to s 2 W and checked that it agrees with the results in the standard quantum field theory textbooks e.g. refs. [50,51]. Furthermore, the constraints we obtain from S, T , and U parameters are in a good agreement with that of ref. [22] which uses the experimental inputs from ref. [26]. We note that our results do not change significantly if we use experimental inputs from ref. [26] instead of ref. [49]. Therefore, in contrast to the claim in refs. [16,17], S, T , and U electroweak precision parameters pose significant constraints on δ as a function of m φ , especially for large m φ .
We now turn to constraints on our parameter space from the collider searches for additional neutral Higgs boson. Once again, for invisible searches, an upper bound is given on µ inv (defined in eq. (3.1)) which translates into for the exotic Higgs φ. For a bound reported on µ inv at each m φ and for a fixed κ inv , we can § Additionally, in eqs. (24) and (25) of ref. [15], we note that there is an additional factor of 1 2 in the term with A 0 (m), but that term drops out of the Peskin-Takeuchi paramaters as it is independent of p 2 .      Table 5.1), along with the strongest constraints from the precision probes for h → γγ (red) and h → W W → eνµν (blue). Also shown are the constraints from the Peskin-Takeuchi parameters for the SM extended with a real SM gauge-singlet scalar Higgs (gold). The 3D plots here showcase various bounds on all the free parameters of this model, and more details are shown in Figure 5.2 with some 2D slices with fixed κ inv .
then solve for the allowed values of δ using the above equation.
For searches for additional Higgs boson in j th SM final state, an upper limit is reported on For φ boson, this limit can be recasted, using eq. (4.21), as from which we can extract the allowed values of δ for a fixed κ inv for each m φ . Various collider searches that look for (additional) neutral Higgs boson(s) in various decay channels (including invisible searches) that pose significant constraints on the parameter space of (δ, κ inv , m φ ) are listed in Table 5.1. The relevant data associated with these searches was obtained, in part, from the files provided with the publicly available Fortran code Higgs-Bounds5 [63]. And the widths and branching ratios of the SM Higgs boson for various masses are obtained using the program HDECAY [64,65]. In Figure 5 regions with dashed borders are the bounds from the invisible searches for φ. The labels for each of these shaded regions correspond to the ones listed in Table 5.1. Gold dashed lines show the constraints from electroweak precision observables, namely the Peskin-Takeuchi S, T , and U parameters. And, dotted lines show the bounds from the precision probes for h → W W → eνµν and h → γγ which typically are much more constraining than the searches for φ. The 3D plots in Figure 5.3 show the richness of various bounds from the searches for neutral Higgs bosons in various decay channels on the parameter space of (δ, κ inv , m φ ). Various 2D slices of the 3D plots in Figure 5.3 with fixed κ inv were shown in Figure 5.2.
In Figure 5.4, we show some future projections for δ as a function of the real singlet scalar mass m φ for κ inv = 0, 1. Here we only show the sensitivities of ILC for φ from the φ → bb decays and using a model-independent recoil mass method in e + e − → Zφ process [66][67][68]. The solid and dotted lines show the present collider constraints from the searches for φ and precision probes for h respectively (see Figures 3.1 and 5.2). The violet dashed lines show the projected sensitivity of the ILC with √ s = 250 GeV for φ → bb searches [66] with 0.5 ab −1 and beam polarization of (80, 30)% for (e − , e + ) respectively. The orange dashed lines show the expected sensitivity, obtained using the model-independent recoil mass method in e + e − → Zφ process, of the 250 GeV ILC with integrated luminosity of (0.9, 0.9, 0.1, 0.1) ab −1 for (e − L e + R , e − R e + L , e − L e + L , e − R e + R ) respectively and beam polarization of (80, 30)% for (e − , e + ) respectively [67]. And, the green dashed lines show similar results but with 500 GeV ILC with integrated luminosity of (1.6, 1.6, 0.4, 0.4) ab −1 for (e − L e + R , e − R e + L , e − L e + L , e − R e + R ) respectively [67]. Both orange and green dashed lines are independent of the choice for κ inv as the mass of φ can be measured using the recoil mass against the Z boson (reconstructed from µ + µ − ) independent of the decays of φ. The dash-dotted lines in the right panel with κ inv = 1 are the future projections for invisible searches of the observed Higgs boson h at HL-LHC and 250 GeV ILC (see Figure 3.1) that indirectly constrain the parameter space.
To summarize, we recasted the bounds/projections reported in various collider searches/studies, for neutral Higgs boson(s), for the depletion factors (δ, κ inv ) of the already discovered 125 GeV Higgs boson h that occur in the SM extended with a real scalar singlet. And, we found that the precision studies of the observed Higgs boson h often provide for the strongest constraint/reach in the parameter space (δ, κ inv , m φ ) compared to the direct searches for the additional physical Higgs φ, at least for m h /2 ≤ m φ ≤ 2m h . For large m φ , we found that the precision electroweak observables S, T, and U can impose a strong constraint on the parameter space. The results obtained in this section for the real singlet scalar extension suggest that in various other extended Higgs sectors that contain "the depleted Higgs boson", the precision probes for the 125 GeV Higgs boson alone can often best constrain such a class of models.

VI. HIGH MULTIPLICITY OF REAL SINGLET SCALARS
Finally, we also briefly consider an extension to the SM with N real SM gauge-singlet scalars S i=1,2,...,N , each with an invisible width Γ inv , where the gauge basis {h SM , S i } is related to the mass basis {h, φ i } in the following way: Here, for purposes of illustration, we chose that the gauge-singlets S i mix equally among themselves (parameterized by ) and with the SM Higgs h SM (parameterized by η). Note that h here stands for the physical Higgs boson with mass ∼ 125 GeV that is already discovered, and φ i are the exotic Higgs with masses taken to be m h /2 ≤ m φ i ≤ 2m h for simplicity. The orthogonality of the above matrix implies that The gauge eigenstates can be expressed in terms of the mass eigenstates: We once again find that the total widths, branching ratios, and production cross-sections of h are same as in Section II, and for the exotic Higgs bosons φ i , the results for φ from Section IV are directly applicable but with the replacement δ → η = δ √ N , for example: Therefore all the bounds in Section V can be reinterpreted for each φ i by simply rescaling the bounds by √ N . We can then immediately see that the bounds coming from the searches for φ i become weaker as N gets larger, and the strongest constraints often come directly from the precision probes for the 125 GeV Higgs boson h. To illustrate this, we randomly generate m φ i ∈ [m h /2, 2m h ] and choose the strongest bound from the bounds obtained for each m φ i . We then iterate these steps several times, so that it is evident that the strongest bound always (often) comes from the precision probes for the 125 GeV Higgs boson h for large (small) N . Therefore, as remarked at the end of the previous section, we found that the indirect precision probes for the observed Higgs boson with m h ∼ 125 GeV tend to give the most powerful constraint on the depletion factors (δ, κ inv ) in the SM extended with a high multiplicity of real singlet scalars, albeit under some simplifying assumptions. And, the constraints from the direct searches for the additional exotic Higgs states φ i get weaker as the multiplicity or the invisible width of the scalars increase.

VII. CONCLUSION
In this paper, we obtained the latest bounds on the universal depletion factor δ that suppresses all the couplings of the observed Higgs boson h to SM final states from the recent LHC searches in various decay channels as a function of the other depletion factor κ inv , a parameter related to the invisible width of h (see eq. (2.3)). We argued that these bounds indirectly constrain many extended Higgs sectors that give rise to δ and κ inv , and are also in general comparable to or stronger than the direct searches for the additional Higgs states, at least if their masses are in between m h /2 and 2m h .
To demonstrate we considered various bounds that come from the collider searches for a exotic Higgs boson φ that occurs in the SM extended with a real SM gauge-singlet scalar that decays only invisibly to some exotic hidden sector fermions. We also obtained the constraints on δ as a function of the mass of the exotic Higgs from precision electroweak observables (in particular the Peskin-Takeuchi parameters). Although, the constraints from the S, T, and U parameters are not the strongest bounds for m h /2 ≤ m φ ≤ 2m h , these bounds get very strong for higher m φ . And, moreover, we also considered some future sensitivities for the observed Higgs boson h and for the exotic Higgs boson φ, to show that the parameter space of the extended Higgs sectors, that lead to the universal depletion factor δ and an invisible width factor κ inv for h, get more constrained and will eventually measure the deviations from the SM predictions and lead the way to the discovery of the additional Higgs states.
Finally, we also considered the case where there are actually N real singlet scalars, with some simplifying assumptions for illustration purposes, and found that the indirect bounds from the precision probes for the 125 GeV Higgs boson h alone tend to moderately constrain such scenarios. The cases with larger number of such scalars and/or larger invisible widths for h are much more constrained from the indirect searches.