Extracting the mass scale of a second Higgs boson from a deviation in $h(125)$ couplings

We investigate the correlation between a possible deviation in the discovered Higgs boson $h(125)$ couplings from the Standard Model prediction and the mass scale ($M_{\text{2nd}}$) of the next-to-lightest Higgs boson in models with non-minimal Higgs sectors. In particular, we comprehensively study a class of next-to-minimal Higgs sectors which satisfy the electroweak $\rho$ parameter to be one at tree level. We derive an upper limit on $M_{\text{2nd}}$ by imposing bounds from perturbative unitarity, vacuum stability, triviality and electroweak precision data as functions of the deviation in the $hVV$ ($V=W,Z$) couplings. Furthermore, we discuss the complementarity between these bounds and the current LHC data, e.g., by considering direct searches for additional Higgs bosons and indirect constraints arising from the measured $h(125)$ signal strengths.


I. INTRODUCTION
The existence of at least one isospin doublet scalar field is strongly suggested by the discovery of the Higgs boson h(125) at the LHC and the measurement of its properties, which are consistent with those of the Standard Model (SM) Higgs boson [1]. This experimental fact brings us to the natural question whether the observed Higgs boson is unique or it corresponds to one of the resonances of a wider structure. The latter possibility requires the Higgs sector to be extended from the minimal form. On the other hand, a non-minimal shape of the Higgs sector is expected by New Physics (NP) paradigms (e.g., composite Higgs models and supersymmetry) embedded in physics beyond the SM (BSM). Therefore, the detection of a second Higgs boson would be a clear evidence of NP.
The mass of a Higgs boson is one of the most critical parameters for its direct detection at collider experiments. In this paper, we aim to systematically derive the limits on the mass of a second Higgs boson in next-to-minimal renormalizable Higgs sectors, i.e., those composed of one isospin doublet plus an extra Higgs field with a non-vanishing Vacuum Expectation Value (VEV). If we require the VEV of the extra Higgs field not to spoil the relation for the electroweak parameter ρ = 1 at tree level, the simplest three choices are: the Higgs Singlet Model (HSM), the 2 Higgs Doublet Model (2HDM) [2] and the Georgi-Machacek (GM) model [3,4] 1 . These extensions, which are commonly understood as low energy descriptions of underlying NP scenarios, have been largely considered in the literature because of their connection with several open questions of the SM. It is known that the HSM can provide a candidate for dark matter [5], while models with Higgs triplets are involved in generating neutrino masses through the so-called type II seesaw mechanism [6]. Conversely, the interest in 2HDMs is mainly motivated by supersymmetric extensions of the SM. Furthermore, the reason to consider models with an extended Higgs sector is phenomenological. As mentioned above, the discovery of a second Higgs boson would necessarily require to build-up a nonminimal structure for the Higgs sector. In addition, non-minimal Higgs sectors can be indirectly probed by precise measurements of the h couplings to SM particles, because the h state is there generally realized through a non-zero mixing among all the other scalars with the same quantum numbers, resulting deviations in the h couplings from the SM prediction.
Regarding, e.g., the coupling to a vector boson pair, the current 1σ uncertainty at the LHC is indeed about 10% [1], so that there is still room for a sensible NP contribution. Such uncertainty is expected to be reduced to ∼ 5% by the forthcoming LHC High Luminosity option [7,8] and even to 0.5% at future e + e − colliders [9].
Basically, the masses of extra Higgs bosons within a given model are free parameters.
However, it is possible to extract their order of magnitude by taking into account theoretical issues within particular BSM scenarios. For example, it is known that perturbative unitarity constrains the size of dimensionless quartic couplings in the Higgs potential which actually enter the expression of the physical Higgs boson masses. Originally, this method was applied to set an upper limit on the SM Higgs boson mass by Lee, Quigg and Thacker [10].
Afterwards, the same technique was carried out in various extended Higgs sectors such as 2HDMs [11][12][13][14]. Besides that, the reliability of a perturbative approach requires the scalar potential to be bounded from below in any direction of the field space. Such requirement is usually referred to as vacuum stability and it provides further constraints on the parameter space of non-minimal Higgs sectors. Furthermore, one requires the absence of Landau poles up to a certain cutoff of a given model: this is the so-called triviality constraint.
Bounds on the mass of extra Higgs bosons can be extracted by considering experimental issues as well. We here take into account the constraints coming from the Electroweak Precision Tests (EWPTs) and the currently available LHC data, which are based both on the null excess of signatures from direct searches for extra Higgs bosons and on the analysis of the h(125) signal strengths. Combining both theoretical and experimental requirements, we restrict the possible allowed values of the extra Higgs masses, depending on the model and its parameter configurations.
It is useful for this analysis to discuss the decoupling and the alignment limits of nonminimal Higgs sectors. The decoupling limit is defined in such a way that all the masses of the extra Higgs bosons are taken to infinity, and eventually only the h state remains light.
In this limit, the extended Higgs sectors effectively reduce to the minimal one and all the observables relevant to h, such as the couplings to SM particles, do not deviate from the SM prediction. On the other hand, the alignment limit is defined such that the h state and the Nambu-Goldstone (NG) bosons emerging from the electroweak spontaneous symmetry breaking fill the same doublet field, which carries the whole VEV v, fixed by v = ( √ HSM Φ(2, 1/2) and S(1, 0) 2HDM Φ 1 (2, 1/2) and Φ 2 (2, 1/2) GM model Φ(2, 1/2), ξ(3, 0) and χ(3, 1) This paper is organized as follows. In Sec. II, we define the three models with non-minimal Higgs sectors, i.e., the HSM, the 2HDM and the GM model. In Sec. III, we numerically extract, for each model, the upper limit on the mass of the next-to-lightest Higgs boson by imposing the theoretical constraints (unitarity, vacuum stability, triviality) and those coming from the EWPTs. In Sec. IV, the complementarity between the bounds from the LHC data and those of Sec. III is studied. Our conclusions are summarized in Sec. V.

II. EXTENDED HIGGS SECTORS
We briefly review the three extended Higgs models considered in this paper, namely, the HSM, the 2HDM and the GM model. The scalar field content is summarized in Table I.
Throughout the paper, we use the shorthand notation: s X = sin X, c X = cos X and t X = tan X for an arbitrary angle parameter X. In each model, the symbol h is used to denote the discovered Higgs boson at the LHC with a mass of 125 GeV (m h = 125 GeV).

A. Higgs Singlet Model
The most general scalar potential in the HSM has the following form: where the doublet and the singlet fields are respectively parameterized as In Eq. (2), G ± and G 0 are the NG bosons which are absorbed by the longitudinal components of the W ± and Z bosons, respectively. The VEV of the singlet field v S contributes neither to the electroweak symmetry breaking nor to the fermion mass generation. As a consequence, we can set v S = 0 without loss of generality because of the shift symmetry of the singlet VEV [15,16], and we will adopt it throughout this paper.
From the tadpole conditions we can eliminate t S and µ 2 parameters. In Eq. (3) and in the following, the symbol | 0 denotes that all the scalar fields are taken to be zero after the derivative. The mass eigenstates for the neutral Higgs bosons can be defined as The squared matrix elements M 2 ij (i, j = 1, 2) in the basis (s, φ) are given in terms of the parameters in the potential as The mass eigenvalues for H S (m H S ) and h (m h ), and the mixing angle α are easily obtained: These relations can be inverted to express each element of the squared mass matrix in terms of the mass eigenvalues: From Eqs. (5) and (6), we see that the decoupling limit is defined by m 2 S → ∞, where m H S and α become infinity and zero, respectively. On the other hand, the alignment limit is obtained by taking µ ΦS → 0, in which the mixing angle α vanishes. In this limit, m H S is not necessarily large. Conversely, from the second equations of (5) and (7) we find that a large value of m H S with non-zero α is realized only by taking a large value of the quartic coupling λ.
Therefore, in the case α = 0, there must be an upper limit on m H S by imposing perturbative unitarity, vacuum stability and triviality bounds (see App. A). We will quantitatively derive such limit in the next section. Following the above discussion, the HSM can be described by 5 independent parameters after fixing the VEV v and m h : The kinetic and Yukawa terms (here and in the following we explicitly write only those for the third generation fermions) are given by where Q 3 L = (t, b) L and L 3 L = (ν τ , τ ) L and Φ c = iτ 2 Φ * . The covariant derivative for Φ reads where g 2 and g 1 are the SU(2) L and U(1) Y gauge couplings, respectively. We see that the singlet field S does not couple to the SM fermions and gauge bosons, so that interaction terms for H S with SM fields are only generated by the mixing. In terms of the mass eigenstates of the Higgs bosons, we thus obtain:

B. 2-Higgs Doublet Model
In order to avoid tree level flavour changing neutral currents in the 2HDM, we impose a discrete Z 2 symmetry [17], which can be softly broken. The Z 2 charge assignment for the two doublets is: The Higgs potential is given by where µ 2 12 is the soft-breaking term of the Z 2 symmetry. In general, the µ 2 12 and λ 5 parameters can be complex, but we assume them to be real for simplicity.
A convenient basis for the Higgs fields is the so-called Higgs basis (Φ, Ψ) [18] which is where In this basis, the VEV v(= v 2 1 + v 2 2 ) and the NG bosons (G ± and G 0 ) belong to the same doublet. Namely, In Eq. (14), H ± and A are the physical singly-charged and CP-odd Higgs bosons, respectively. The two CP-even Higgs states h ′ 1 and h ′ 2 can mix each other. Their mass eigenstates are defined by By imposing the tadpole conditions we can eliminate µ 2 1 and µ 2 2 . The masses for A (m A ) and H ± (m H ± ) are then given by where M 2 = µ 2 12 /(s β c β ). The relation between the CP-even Higgs boson masses and the matrix elements M 2 ij in the (h ′ 1 , h ′ 2 ) basis is given in Eqs. (6) and (7) after the replacement of (α, H S ) → (α − β, H). The squared matrix elements are given in terms of the potential parameters by the following relations: where λ 345 = λ 3 + λ 4 + λ 5 . From the discussion above, the scalar potential can be fully described by the following set of independent parameters (after fixing v and m h ): with 0 < β − α < π.
Let us discuss the decoupling and the alignment limits in the 2HDM. The decoupling limit is given by M 2 → ∞, by which all the masses of H ± , A and H become infinity, and t 2(β−α) becomes zero (equivalently s β−α → 1). On the other hand, the alignment limit is defined by taking s β−α → 1, so that the h ′ 1 state in Eq. (14) corresponds to the h state. Similarly to the HSM, if we take s β−α = 1 the decoupling limit cannot be reached, because a large value for m H is only realized by a large value of the scalar quartic couplings (which are disfavored, e.g., by perturbative unitarity). This is clear from the relation: Regarding the kinetic and Yukawa interaction terms, they are given by where the covariant derivative D µ is the same as Eq. (10). The Φ f (f = t, b, τ ) fields are Φ 1 or Φ 2 depending on the Z 2 -charge assignment for the right handed fermions. For the latter, there are 4 independent choices which lead to 4 different types of Yukawa interactions [19,20], referred to as Type-I, Type-II, Type-X and Type-Y [21]. The combination of (Φ t , Φ b , Φ τ ) is determined in each type as follows: For example, Type-II is realized by setting the charge assignments as (t R , b R , τ R ) → (−t R , +b R , +τ R ) and taking all the left-handed fermions to be Z 2 -even. In terms of the Higgs boson mass eigenstates, we obtain the following interaction terms: where I f = +1/2(−1/2) for f = t(b, τ ) and P L,R are the projection operators for left-/righthanded fermions. The mixing factors ξ f are given by: It is straightforward to check that, in the alignment limit s β−α → 1, all the h couplings coincide with those of the SM Higgs boson at tree level.

C. Georgi-Machacek Model
The Higgs potential in the GM model can be constructed in terms of the SU(2) L ×SU(2) R bi-doublet Φ and bi-triplet ∆ fields: where χ c = C 3 χ * is the charge conjugated χ field with The neutral component fields are expressed by where v φ , v χ and v ξ being the VEV's for φ 0 , χ 0 and ξ 0 , respectively.
Assuming the vacuum alignment: v χ = v ξ (≡ v ∆ ), the VEV ∆ is proportional to the 3 × 3 identity matrix. In this configuration, the global SU(2) L × SU(2) R symmetry is spontaneously broken down to the SU(2) V symmetry, i.e., the so-called custodial symmetry.
The most general scalar potential for the Φ and ∆ fields is given by where τ a and t a (a = 1-3) are the 2 × 2 and 3 × 3 matrix representations of the SU (2) generators, respectively. The unitary matrix P enters in the similarity transformation P (−iǫ a )P † = t a with ǫ a being the adjoint representation of the SU(2) generators. The explicit form of P is given as We note that the custodial symmetry in the Higgs potential is broken by the U(1) Y hypercharge gauge interaction at one-loop level. The effects of the loop induced custodial-breaking terms, which cannot be described in terms of Φ and ∆, were discussed in Ref. [22].
The mass eigenstates of the physical Higgs bosons can be classified in terms of the SU(2) V multiplets, namely, the 5-plet (H ±± 5 , H ± 5 , H 0 5 ), the two 3-plets (H ± 3 , H 0 3 ) and (G ± , G 0 ), and the two singlets H 1 and h. The Higgs bosons belonging to the same SU(2) V multiplet are degenerate in mass. These mass eigenstates are related to the original states given in Eq. (25) by the following transformations: where R ± 5 and R r 5 are the orthogonal 3 × 3 matrices which separate the 5-plet Higgs bosons from the other singly-charged and CP-even scalar states, respectively. Their explicit forms are given by The other two matrices involved in Eq. (30) are where and α describes the mixing of the CP-even singlet states. By using the two independent tadpole conditions we can eliminate the µ 2 Φ and µ 2 ∆ parameters. The mass eigenvalues of the SU(2) V 5-plet and 3-plet Higgs bosons (m H 5 and m H 3 ) are then given by The relation between the masses of the CP-even Higgs bosons and the matrix elements M 2 ij in the basis of (ξ ′ r , φ r ) can be obtained by using Eqs. (6) and (7) after the replacement of H S → H 1 . The matrix elements have the following expressions in terms of the potential parameters: Summarizing, the GM model can be described by 7 independent quantities, after fixing v and m h : Let us now discuss the decoupling and the alignment limits. The decoupling limit is realized by taking t β → ∞ with a negative value of µ 1 . In this limit, m H 3 and m H 5 become infinity as seen by Eq. (34). In addition, among the matrix elements given in Eq. (35), only M 2 11 becomes infinity, so that we obtain m H 1 → ∞ and α → 0. Differently from the HSM and the 2HDM, the alignment limit cannot be taken without the decoupling limit, because t β → ∞ is necessarily required in order to have the doublet field Φ carrying the whole VEV v.
Finally, let us explicitly write the kinetic and Yukawa Lagrangian terms of the GM model: where the covariant derivatives are expressed as The trilinear interaction terms of the physical Higgs bosons with SM fermions and gauge bosons are given by: where There are several remarkable points to be stressed about the interaction terms given in Eq. (40). First, the 5-plet Higgs bosons have a fermiophobic nature, i.e., they do not couple to fermions, but they couple to the SM gauge bosons. Second, the coupling structure of the 3-plet Higgs bosons H 0 3 and H ± 3 is the same as that for A and H ± in the Type-I 2HDM. Finally, the SM-like Higgs boson h coupling to the gauge bosons can be larger than the SM prediction, because of the factor of 2 √ 6/3 as shown in Eq. (41). This is a peculiarity of the GM model, because κ V ≤ 1 in the HSM and the 2HDMs.
In Table II, we summarize the relevant properties of the three extended Higgs sectors here considered. constraints. In addition, we will concern about the compatibility with the experimental values of the S and T parameters introduced by Peskin and Takeuchi [23] which describe the oblique corrections to the electroweak processes. In particular, we require the predictions of ∆S ≡ S NP − S SM and ∆T ≡ T NP − T SM to be within the 95% CL region of the ∆χ 2 fit ∆S exp = 0.05 ± 0.09, ∆T exp = 0.08 ± 0.07, with the correlation factor ρ ST being 0.91. The analytic formulae for ∆S and ∆T are given in Appendix C for each of the extended Higgs models.
We introduce M 2nd as the mass of the next-to-lightest Higgs boson, assuming the lightest one to be the discovered Higgs boson h. By definition, M 2nd corresponds to m H S in the HSM, while in the 2HDM and the GM model, it is defined by: In the following analysis, we enforce the bounds in two steps denoted by "Bound A" and "Bound B". First, we impose the tree level perturbative unitarity, the vacuum stability and the constraint from the S and T parameters (Bound A). In addition to the constraints of Bound A, we then impose the improved vacuum stability and the triviality constraints We can see that the extracted limit on M 2nd in the 2HDM (typically less than 1 TeV) is much smaller than that given in the HSM (typically a few TeV). We note that this result does not depend on the type of Yukawa interaction. In Fig. 3 so we here consider both values of α. In Fig. 4, we show the maximally allowed value of M 2nd as a function of κ V by imposing Bound A for t β = 1, 2 and 3 (left panel) and 5, 10 and 20 (right panel). As we mentioned in Sec. II C, κ V > 1 is allowed and its maximal value is found at t α = −2 √ 6/(3t β ). For example, for t β = 3(5), we obtain Max(κ V ) ≃ 1.08(1.03).
For this reason, there is an end point in the curves shown on the right panel. We find that the maximal allowed value of M 2nd monotonically increases as κ V is getting large. However, as long as we take a finite value of t β , there is always an upper limit on M 2nd , i.e., the decoupling limit cannot be taken unless t β → ∞ (corresponding to the alignment limit).
In Fig. 5, we show the contour plots for M 2nd on the κ V -t β plane by imposing Bound A (left) and by imposing Bound B with Λ = 10 4 GeV (center) and 10 8 GeV (right). The white regions are not allowed due to the existence of a maximal value for κ V . We can see that the maximal value of M 2nd is smoothly getting large when t β and/or κ V becomes large.
As compared to the case of the 2HDMs, M 2nd can be typically above 1 TeV when we take a large value of t β . For example, for κ V < 0.99, we obtain the maximal allowed value of M 2nd to be about 1 (2) TeV in the case of t β = 5(10). If we impose Bound B, the maximal M 2nd becomes smaller, but still M 2nd > 1 TeV is allowed for κ V < 0.99 with Λ = 10 4 GeV.

IV. COMPLEMENTARITY BETWEEN EXPERIMENTAL AND THEORETICAL BOUNDS
In this section, we discuss the complementarity between the bounds discussed in Sec. III and those from the LHC data. For the former, we impose Bound A which does not depend on the cutoff Λ. For the latter, we take into account the bounds from direct searches for additional Higgs bosons and from the data for the discovered Higgs boson h(125). As we saw in the previous section, Bound A provides an upper limit on M 2nd depending on the value of κ V . Conversely, bounds from LHC data typically provide a lower limit on the mass of the extra Higgs boson. Therefore, by combining these two types of bounds, we can further narrow down the possible allowed region for M 2nd for each extended Higgs model.
Here, we consider the masses of extra Higgs bosons to be above 350 GeV, i.e., beyond the threshold of the decay of neutral Higgs bosons into tt. For the 2HDM and the GM model, we take t β ≥ 1, because the case with t β < 1 is highly disfavored by the various B physics experiments [25,26].
Let us discuss the bounds from direct searches. In the aforementioned scenarios, the main decay modes of the extra Higgs bosons are the following: H → tt/V V /hh, A → tt/Zh, H ± → tb/W ± h (in the 2HDM), However, as we will show below, in a large t β scenario i.e., t β = O(10), the production cross section of the extra Higgs bosons is highly suppressed, so that the constraint on the mass of the extra Higgs boson becomes weak regardless of the type of Yukawa interaction 3 .
First, let us discuss the searches for additional neutral Higgs bosons (H). We here consider the following processes: We note that, although there are other production modes for H such as the vector boson fusion and the vector boson associated processes, the cross section of these modes are negligibly small in the scenario with κ V ∼ 1. Thus, we only take into account the gluon fusion production in this analysis. Regarding the top quark associated production, the search for pp → ttH followed by H → tt decay has been performed at the LHC using the data of 13 TeV and 13.2 fb −1 [29]. The current bound assuming 100% of the BR(H → tt) is, however, not so stringent. For example in the 2HDMs (regardless the type of Yukawa interactions), no bound on the mass of H has been taken when tan β 0.3. Therefore, the top quark associated process will be neglected in this analysis 4 . The processes (i)-(iv) have been searched for using the 8 TeV data with the integrated luminosity of 20.3 fb −1 in Refs. [30], [31], [32] and [33], respectively. Due to the fact that there is no significant excess in the number of signal events with respect to that given in the SM, the 95% CL upper limit on the cross section times the branching ratio has been provided for each process. Concerning the process (ii), 2 In the Type-X 2HDM, H ± → tb can also be replaced by H ± → τ ± ν for t β 10, s β−α = 1, and m Φ = M = 500 GeV. This is not the case for the other 2HDM types. 3 In the LHC Run-2 and the High-Luminosity LHC experiments, the large t β scenario can also be constrained [27,28] via the pair production of the extra Higgs bosons, whose cross section does not depend on t β . 4 In the HSM, the production cross section of pp → ttH S is proportional to s 2 α , so that the search for H S → tt is less important with respect to the 2HDM as long as we take s α ≪ 1. In the GM model, the properties of H 1 and H 0 3 are quite similar to those of H and A in the Type-I 2HDM, respectively, so that we can apply the similar phenomenological analysis of the Type-I 2HDM for these neutral Higgs bosons. the hh → γγbb, τ τ bb, W W * bb and bbbb decay modes were independently analysed [31], and similarly for the process (iv), for which the h → bb and h → τ τ modes were analysed [33].
In order to compare the bounds driven by the experiments with the corresponding theory predictions, let us evaluate the cross section for the gluon fusion production for H (σ ggH ) by using the following approximation: where σ ggh SM is the gluon fusion cross section in the SM and Γ(X → gg) is the decay rate for the X → gg mode. The reference values of σ ggh SM at 8 TeV are given in [34].
Second, let us discuss direct searches for singly-charged Higgs bosons. Typical main decay modes are given in Eqs. (45) and (46) in the 2HDM and the GM model, respectively. The search for charged Higgs bosons decaying into the tb mode has been surveyed in Ref. [29].
The current bound is not so stringent. For example, in the Type-II 2HDM no bound has been taken on the mass of H ± when t β 0.5, which is also valid for all the other types of ) for an arbitrary value of v ∆ can be extracted as follows: where v Ref.
∆ is a reference value of v ∆ which is set to be 16, 25 and 35 GeV in Ref. [37]. The ∆ are also presented in Ref. [37].
Apart from the direct searches for extra Higgs bosons, we need to consider also the constraint on the parameter space from the h(125) data at the LHC. Here, we take into account the signal strengths µ X for X = γγ, ZZ, W W and τ τ defined by The measured values of µ X are given by the combined analysis of the ATLAS and CMS experiments using the LHC Run-1 data [1] as follows: and we shall require each prediction for µ X to lie within the 95% CL region.
In Fig. 6, we show the allowed parameter space on the κ V -m H S plane in the HSM. The region above the black solid curve is excluded at 95% CL by the S and T parameters, while the blue shaded region is excluded at 95% CL by the direct search for the gg → H S → ZZ process, which turns out to set the most stringent constraint among the direct search and signal strength data. However, we see that the region excluded by the direct searches is almost ruled out by the constraint from the S, T parameters, so that the LHC data do not significantly improve our bounds for m H S with respect to Sec. III.
Next, the constraints on the parameter space of the 2HDM are shown in Figs. 7 and 8. In Fig. 7, we show the excluded region at 95% CL by the direct searches (shaded region) and µ X (indicated by the green dashed line) on the We also display the corresponding value of κ V on the top horizontal axis. The results in the Type-I, -II, -X and -Y 2HDMs are shown from the top to bottom panels, while those for t β = 1, 2 and 3 are shown from the left to right panels. In all the plots, we take M = m Φ .
Regardless of the type of Yukawa interactions, the bound from the direct searches becomes milder when we take a larger value of t β , because the top quark loop contribution to the gluon fusion cross section is suppressed by a factor ∝ cot 2 β. In fact, if we take t β 10, almost no region on the c β−α -m Φ plane shown in this figure is excluded by the direct searches.
Concerning the bound from µ X , they give a severe constraint on κ V particularly in the Type-II and Type-Y 2HDMs, by which |κ V − 1| larger than 1% are not allowed. This constraint tends to get stronger when we take a larger value of t β except for the Type-I 2HDM. Let us now comment on the case with M = m Φ . For m Φ < M, the constraint from gg → H → hh tends to be milder, because the branching ratio of H → hh becomes small. On the contrary, the constraint from gg → H → ZZ tends to be slightly stronger due to a little enhancement of the branching ratio of H → ZZ. In any case, the total excluded region with M < m Φ is found not to change so much with respect to our initial assumption. On the contrary, the case M > m Φ is totally disfavored by the vacuum stability bound.
Let us combine the bounds from the LHC data discussed above and Bound A in the 2HDM. In Fig. 8, the black (red) dots are allowed by only Bound A (both Bound A and the LHC data). We here keep the assumption of the degeneracy in mass of the extra Higgs bosons, and we scan over the value of M . The results for Type-I and Type-II 2HDMs are shown on the left and right panels, respectively, and the value of t β is taken to be 1, 2, 3 and 10 from the top to bottom panels. We note that the results for the Type-X and Type-Y 2HDMs are almost the same as those for the Type-I and Type-II 2HDMs, respectively. We can see that for t β = 1 (top panels), m Φ 600 GeV is excluded, because of the gg → H/A → tt process (according to Fig. 7). For larger t β , the region filled by the black and red dots is getting the same in the Type-I 2HDM, namely, the bounds from the direct search and µ X become less important. In contrast, in the Type-II 2HDM, the region filled by the red dots is much smaller than that filled by the black dots even for the case with large t β , because of the constraint from µ X . In conclusion, we find that for smaller The parameters m H 1 , µ 1 and µ 2 are scanned (m H 1 ≥ 350 GeV) with a large enough range to maximize the allowed parameter space. We note that the case with t β 3 is excluded by the direct searches for H ±± 5 → W ± W ± up to m H 5 to be 800 GeV [37]. We see that a larger value of m H 5 is allowed in the case with κ V > 1 by both the constraints from Bound A and the LHC data. For larger t β , the allowed range of κ V is getting smaller. For example, we  Table III for reference values of ∆κ V , assuming the cutoff to be above 10 TeV under Bound B. In addition, we have discussed the complementarity between the non-LHC and the LHC bounds, i.e., the direct searches and the h(125) signal strengths.
In the HSM, the LHC data do not improve the non-LHC bounds for κ V 0.95. On the  contrary, in the 2HDMs, the constraint from the LHC data can be important depending on the type of Yukawa interaction and the value of t β . For the Type-I and Type-X 2HDMs with t β ∼ 1, the LHC data restrict κ V 0.98 and the mass of extra Higgs bosons to be above ∼ 600 GeV. For larger t β , the LHC data are less effective and they provide no further constraint with respect to the non-LHC bounds. For the Type-II and Type-Y 2HDMs, the µ X data constrain κ V 0.99 for t β ∼ 1, and such bound becomes stronger for larger t β .
Finally, in the GM model, we found that the case with κ V 1 is favored by both the LHC and non-LHC bounds. In addition, for a larger value of t β , the allowed range of κ V is getting smaller. For example, if we require m H 5 (= m H 3 ) ≥ 500 GeV, we obtain 0.96 κ V 1.03 and 0.99 κ V 1.01 for t β = 5 and 10, respectively.
In conclusion, from the analysis performed in this paper, we have clarified the connection between the scale of the second Higgs boson mass in the non-minimal Higgs sectors here considered and the deviation in the hV V couplings from the SM prediction. Since the hV V couplings will be precisely measured at future collider experiments such as the High-Luminosity LHC (a few percent level) and e + e − colliders (less than 1% level at the International Linear Collider [38] with 500 GeV of the collision energy [9]), we can obtain precise information of the mass of the next-to-lightest Higgs boson even without its discovery at the LHC.

Appendix A: Theoretical constraints
We briefly review the theoretical constraints, i.e., the perturbative unitarity, the vacuum stability and the triviality we enforce in the models with an extended Higgs sector. We then present relevant analytic expressions for these constraints.
We first discuss the bound from the perturbative unitarity. The request of the S-matrix unitarity for 2 body to 2 body elastic scattering processes for scalar bosons, assuming the validity of perturbative calculations, leads to the following condition: where a J are partial wave amplitudes with total angular momentum J. For the purpose of this paper, we define the perturbative unitarity bound by where x i are the eigenvalues of the S-wave (J = 0) amplitude matrix, because they give the most stringent constraints.
As stated above, we impose |x i | ≤ 1/2 and derive the allowed regions in the parameter space of each model.
Next, let us discuss the vacuum stability bound. We require that the scalar potential is bounded from below in any direction with large field values. This can be simply expressed by is the scalar quartic part of the potential. The sufficient and necessary conditions to satisfy the vacuum stability constraint in the HSM [5,43] are given by λ ≥ 0, λ S ≥ 0, 2 λλ S + λ ΦS ≥ 0.
The ∆S parameter in the GM model is given by: