One-loop renormalized Higgs vertices in Georgi-Machacek model

We compute renormalized vertices of the 125 GeV Higgs boson $h$ with the weak gauge bosons ($hVV$), fermions ($hf\bar{f}$) and itself ($hhh$) in the Georgi-Machacek model at one-loop level. The renormalization is performed based on the on-shell scheme with the use of the minimal subtraction scheme only for the $hhh$ vertex. We explicitly show the gauge dependence in the counterterms of the scalar mixing parameters in the general $R_\xi$ gauge, and that the dependence can be removed by using the pinch technique in physical scattering processes. We then discuss the possible allowed deviations in these one-loop corrected Higgs couplings from the standard model predictions by scanning model parameters under the constraints of perturbative unitarity and vacuum stability as well as those from experimental data.


I. INTRODUCTION
Discovery of the 125 GeV Higgs boson at the CERN Large Hadron Collider (LHC) had completed the particle spectrum of the Standard Model (SM). This, however, does not necessarily mean that the SM is the ultimate theory describing elementary particle physics, because of theoretically unsatisfactory issues, such as the gauge hierarchy problem and unexplained phenomena of neutrino mass, dark matter and baryon asymmetry of the Universe.
These problems are expected to be solved in new physics (NP) beyond the SM at or above the TeV scale. In NP models, the Higgs sector is often extended from the minimal form assumed in the SM, and its properties strongly depend on the NP scenario. Therefore, a determination of the structure of the Higgs sector using experimental data is important to narrow down possible NP models.
There are basically two ways to identify an extended Higgs sector: the direct search and the indirect search. The former approach is to discover additional Higgs bosons, while the latter is to find deviations in various observables related to the discovered Higgs boson (h) from the SM predictions. So far, no additional Higgs boson has been discovered at the LHC, and this situation makes the indirect search attractive. Currently the Higgs boson couplings are measured with insufficient accuracies at the LHC, e.g., a 10% level uncertainty in the hV V (V = W ± , Z) couplings [1]. They are expected to be measured with much better accuracies in future collider experiments, such as the high-luminosity LHC and e + e − colliders, where they can be determined to the percent or sub-percent level [2,3].
In order to make a sensible comparison with such precision measurements, one needs to reduce theoretical uncertainties in models with an extended Higgs sector. In particular, radiative corrections to the Higgs boson couplings should be taken into account. One-loop corrections to various Higgs boson couplings have been studied in several models with a simple Higgs extension, e.g., models with an additional isospin singlet scalar field, the Higgs singlet model (HSM) [4][5][6][7]; a doublet scalar field, two-Higgs doublet models (THDMs) [8][9][10][11], and a complex triplet field, the Higgs triplet model (HTM) [12,13]. Recently, a numerical tool H-COUP [14] has been constructed to compute various h couplings at one-loop level in the HSM and THDMs without any gauge dependence [15].
In this paper, we investigate one-loop corrections to the Higgs boson couplings in the Georgi-Machacek (GM) model [16,17] which has the capacity of providing Majorana mass factors for hV V , hff and hhh couplings normalized to their SM predictions and discuss their momentum dependence. Section VI discusses and lists theoretical and experimental constraints to be imposed in the parameter scan of the model. Section VII presents the numerical result for the renormalized scale factors by scanning model parameters under the both theoretical and experimental constraints. Section VIII summarizes our findings in this work. Appendices A and B give explicit formulas of the masses and interactions of the Higgs bosons in the model, respectively. In Appendix C, the loop functions are defined, and explicit formulae for contributions from 1PI diagrams that appear in our calculations are presented.

II. THE MODEL
The scalar sector of the GM model is composed of a weak isospin doublet field φ with hypercharge Y = 1/2 and weak isospin triplet fields χ and ξ with Y = 1 and Y = 0, respectively. These scalar fields can be expressed in the SU (2) L × SU (2) R bi-doublet (Φ) and bi-triplet (∆) forms as: where φ c = iτ 2 φ * and χ c = C 3 χ * are the charge-conjugated φ and χ fields, respectively. The matrix C 3 is given by The neutral components are parameterized by where v φ , v χ and v ξ are the VEVs of φ 0 , χ 0 and ξ 0 , respectively. For later convenience, we re-express the two triplet VEVs by The ν parameter describes the deviation from alignment in the triplet VEVs, i.e., ∆ = v ∆ 1 3×3 .
In the following subsections, we first discuss the scalar potential and explain the necessity of introducing SU (2) L ×SU (2) R breaking terms in order to make the model consistent at loop levels. We then give the Lagrangian of the scalar kinetic terms and the Yukawa interactions.
Finally, we discuss the decoupling property of the GM model.

A. Scalar potential
The SU (2) L × U (1) Y gauge-invariant scalar potential can be expressed as follows: where V cust and V cust are respectively given as a function of {Φ, ∆} and {φ, χ, ξ} 2 . V cust is defined such that when it is vanishing, the potential has the most general global SU (2) L × SU (2) R symmetry which is spontaneously broken down to the diagonal part SU (2) V , the so-called custodial symmetry, under the assumption of vacuum alignment: v χ = v ξ or, equivalently, ν = 0. In this configuration, the electroweak rho parameter ρ is predicted to be 1 at tree level as we will see in the next subsection.
Nonetheless, even if we take V cust = 0 at tree level, V cust generally re-appears at loop levels due to, e.g., hypercharge gauge boson loops as a consequence of SU (2) L × SU (2) R breaking effects in the kinetic term. In addition, such loop contributions contain ultra-violet (UV) divergences which cannot be cancelled by counterterms associated with the V cust part alone. Therefore, in order to make the model consistent at loop levels, we need to introduce custodial symmetry breaking terms from the beginning. The simplest choice to make our calculations of renormalized vertices for the discovered Higgs boson consistent is to introduce where ξ = (ξ + , ξ 0 , −ξ − ) T . The other possible terms for V cust can be important for the 2 Given the scalar fields in the model, the most general SU (2) L × U (1) Y gauge-invariant scalar potential has 14 real and 2 complex parameters. Imposing the global SU (2) L × SU (2) R symmetry renders relations among the parameters and results in the custodial symmetric potential given in Eq. (7) described by 9 real parameters, as shown in Ref. [28].
computation of one-loop corrections to physical quantities related to the extra Higgs bosons, but are not our concerns here.
Explicitly, the most general custodial symmetric potential is given by where τ a /2 and t a (a = 1, 2, 3) are the 2×2 and 3×3 representations of the SU (2) generators, respectively. The matrix P gives the similarity transformation P (−i a )P † = t a with a being the adjoint representation of the SU (2) generators, and is given as To obtain the mass eigenvalues for the physical Higgs bosons, one imposes the tadpole conditions at tree level: Using the above three equations, one can re-write the three mass parameters m 2 Φ , m 2 ∆ and m 2 ξ in terms of the other parameters in the scalar potential. We note that in the limit of ν = 0, m 2 ξ also vanishes and the tadpole conditions for χ r and ξ r become identical as a consequence of restoring the custodial symmetry at tree level. Detailed analytic expressions for the physical Higgs bosons and their squared masses are presented in Appendix A for the general ν = 0 case.
We here highlight some important properties of the mass spectrum in the ν = 0 limit.
Therefore, the rotation matrices to separate the Nambu-Goldstone (NG) bosons from the physical CP-odd and singly-charged Higgs bosons become the same. This also shows the recovery of the custodial symmetry. Consequently, all the potential parameters can be expressed in terms of the following 9 parameters:

B. Kinetic terms
The kinetic terms of the scalar fields are given by where the covariant derivatives The weak gauge boson masses are calculated to be whereν ≡ 2 ν(2v ∆ + ν) and g Z ≡ g 2 + g 2 . As in the SM, the electroweak symmetry breaking SU (2) L ×U (1) Y → U (1) EM forces the following relation among the gauge couplings: where α EM is the fine structure constant and c W (s W ) is the cosine (sine) of the weak mixing angle θ W . Using these relations, we can also write g Z = g/c W . From Eq. (14), we can identify the VEV v, which is related to the Fermi's decay constant The tree-level rho parameter is then given by Therefore, a non-zero ν would make the rho parameter deviate from unity at tree level.
This implies that unlike in the SM, the electroweak sector is now empirically fixed by four independent parameters, e.g., the set of {m W , m Z , α EM , ν}. In fact, the necessity of four input parameters in the electroweak sector generally appears in models with ρ tree = 1 [29,30].
In terms of these four parameters, s 2 W and v 2 are given by: as seen in Eq. (B2). This property is not seen in the corresponding couplings of h and H 1 .

C. Yukawa interactions
The Yukawa Lagrangian for the third-generation fermions is given by where Q 3 L = (t, b) T L and L 3 L = (ν τ , τ ) T L . The Yukawa interactions for the other SM fermions can be simply obtained by generalizing the above Yukawa couplings to 3×3 Yukawa matrices.
In the ν → 0 limit, fermion masses are obtained as m f = y f vs β / √ 2 for f ∈ {t, b, τ }. We note that there is another type of Yukawa interaction terms for the χ field, which is expressed as and gives Majorana mass to left-handed neutrinos. Typically, the size of this Yukawa coupling is expected to be as small as O(10 −9 -10 −10 ) for v ∆ = O(1) GeV to reproduce the observed neutrino oscillations. Thus, these interactions do not play any important role in the following discussions and are ignored throughout this paper.
The interaction terms for the physical Higgs bosons are given in Appendix B in the ν → 0 limit. It should be noted that the 5-plet Higgs bosons do not couple to fermions and are thus fermio-phobic, while the 3-plet Higgs bosons do. In fact, the structure of the Yukawa couplings of the 3-plet Higgs bosons is the same as that of the CP-odd and charged Higgs bosons in the Type-I THDM.

D. Decoupling limit
In this subsection, we briefly discuss the decoupling limit in the GM model. As clarified in Ref. [32], the decoupling limit can be realized by taking m ∆ to infinity, with all the extra Higgs boson masses also going to infinity and the SM predictions being reproduced.
In order to clearly see how the decoupling limit works, we expand physical parameters in the decoupling regime (i.e., m ∆ v) in powers of 1/m 2 ∆ . The masses of extra Higgs bosons are expanded as With these mass parameters growing virtually linearly with m ∆ , these extra Higgs bosons are decoupled from the theory in the m ∆ → ∞ limit. It is also seen that the differences among these mass parameters are suppressed by O(1/m ∆ ) or higher. Keeping terms up to order 1/m ∆ , we obtain a relation among these mass parameters [33] On the other hand, the mass of the SM-like Higgs boson h is mainly given by the λ 1 term as in the SM: Next, we check the decoupling behavior of the couplings associated with the SM-like Higgs boson h. At tree level, the h couplings are modified from the SM predictions due to the mixing between the CP-even Higgs bosons and the VEVs. The former and the latter are respectively parameterized by α and v ∆ (or β). Expanding in powers of 1/m 2 ∆ , we obtain As expected, both of these parameters approach zero in the limit of m ∆ → ∞ with µ 1 taken to be finite. The decoupling behavior of the h couplings can be shown more directly by expanding the normalized hV V (c hV V ), hf f (c hf f ) and hhh (c hhh ) couplings by their SM values as As expected, they all become one in the decoupling limit.

III. RENORMALIZATION
In this section, we discuss the renormalization prescription to obtain finite one-loop corrected Higgs boson couplings. Our renormalization is based on the on-shell scheme, where counterterms are introduced to cancel the radiative corrections to the mass parameters (as well as wave functions) for various fields on their mass shells.
In our calculation, unrenormalized one-loop contributions to 2-point and 3-point functions are constructed in the so-called tadpole scheme [34,35] as where A, B, and C refer to particles on the external legs. The first and second terms on the right-hand sides denote the contributions from 1-particle irreducible (1PI) and tadpole inserted diagrams, respectively. Obviously, there is no momentum dependence in the tadpole inserted contributions (Π Tad AB and Γ Tad ABC ). For later convenience, we define the derivative for a generic 2-point function. We assume that effects of custodial SU (2) V symmetry breaking are introduced at the one-loop level; namely, we take the SU (2) V breaking parameter ν = 0 at tree level. Therefore, in the calculations of one-loop diagrams, we can make use of the tree-level properties discussed in the previous section, such as a degenerate mass for the Higgs bosons belonging to the same SU (2) V multiplet, because including deviations from the tree-level properties would be of higher order corrections.
In the subsequent subsections, we discuss the renormalization of the parameters in the gauge sector, the fermion sector and the scalar sector in order.

A. Gauge sector
We shift the following electroweak parameters and the field wave functions of SU (2) L and U (1) Y gauge bosons denoted by W a µ (a = 1, 2, 3) and B µ as: in which we have introduced 6 counterterms. Using Eq. (18), the counterterms δv and δs 2 W are given by Furthermore, it is convenient to define the following counterterms for the wave functions of the physical Z boson and photon fields: The renormalized gauge boson 2-point functionsΠ XY , (XY = W W, ZZ, Zγ, γγ) can then be defined as follows: with Π XY being the nurenormalized 2-point functions defined in Eq. (26).
In order to determine the counterterms in Eq. (28), we impose the following five on-shell conditions, which are the same as those used in the SM [36]: whereΓ γee µ is the renormalized photon-electron-positron vertex. From them, the five counterterms are determined as follows: Using Eq. (30), one then finds As explained in Section II, there is one additional counterterm δν in the GM model. Therefore, we need another condition to fix it. Following the earlier work in Ref. [25], we demand that the electroweak oblique T parameter, T ≡ T GM − T SM with T GM and T SM being respectively the T parameter calculated in the GM model and the SM, be equal to its experimental value: where We will set T exp = 0 in the discussion of numerical analyses.

B. Fermion sector
The renormalization for the fermion sector can be done in the same way as in the SM.
Left-handed and right-handed fermions (ψ L and ψ R ) and their masses m f are shifted as Following Ref. [36], these counterterms are given by where Π f f,V and Π f f,S are the vector and scalar parts of the fermion 2-point functions defined in Eq. (C21) at the one-loop level, respectively. Although another independent wave function renormalization factor δZ f A = (δZ f L −δZ f R )/2 can be constructed, it does not appear in subsequent discussions.

C. Scalar sector
Finally, we discuss the renormalization of parameters in the scalar potential. In particular, we concentrate on the neutral scalar part, because the charged scalar states are not relevant for the discussions of the renormalized Higgs boson vertices in Section V. We shift the parameters defined in Eq. (11) as follows: where the shifts for v and ν are already done in Section III A. We here also shift the mixing angles α 1 and α 2 which become zero at tree level due to the custodial symmetry 3 . We note that there are also counterterms for the three tadpoles of φ r , χ r and ξ r . But these counterterms should be zero in the tadpole scheme [35], as their contributions are already included in the tadpole inserted diagrams in Eq. (26). The wave functions for the CP-odd and CP-even Higgs bosons are then shifted as follows: where δZ ij = δZ ji .
The renormalized 2-point functions for the neutral scalar fields are given bŷ . In addition, δθ SS is δα, δα 1 , δα 2 and δβ for (S, S ) = (H 1 , h), (h, H 0 5 ), (H 1 , H 0 5 ) and (G 0 , H 0 3 ), respectively, and δθ SS = −δθ S S . To determine these counterterms, we impose the following on-shell conditions: Counterterms are then determined as and There are still two counterterms δµ 1,2 that are not fixed by the above conditions. These counterterms appear in the renormalized hhh vertex, and we will discuss how to determine these counterterms in Section V C.

IV. GAUGE DEPENDENCE
In the previous section, we have determined all the counterterms by imposing the on-shell renormalization conditions except for δµ 1,2 . As a result, they can be expressed in terms of 2-point functions defined in Eq. (26). However, it has been known that there remains gauge dependence in the counterterms for the mixing angles, e.g., δβ and δα, in the on-shell scheme as it can be proved using the Nielsen identity [37].
In this section, we first show the gauge dependence in the scalar 2-point functions, particularly for the CP-even and CP-odd scalar bosons. In order to manifestly show the gauge dependence, we perform the calculation in the general R ξ gauge, where the propagator of a gauge boson V (= W, Z) is expressed using the gauge parameter ξ V as We then discuss how one can remove such gauge dependence by employing the pinch technique [38,39].

A. CP-even part
First, we show explicitly the cancellation of the gauge dependence in the mixing of CPeven Higgs bosons. Here, we only show the ξ W -dependent part because the ξ Z part can be simply obtained by the replacements of (g, W, G ± ) → (g Z /2, Z, G 0 ). The 2-point functions where the first term on the right-hand side corresponds to the result calculated in the 't Hooft-Feynman gauge. On the other hand, the second term in Eq. (47) depends on the gauge parameter and is explicitly given by where we have introduced The function B 0 is the Passarino-Veltman's scalar 2-point function [40] defined in Section C 1.
We see that for vanishes at q 2 = m 2 φ as it is expected by the Nielsen identity, so that the counterterms for the mass parameters (δm 2 h and δm 2 and thus the gauge dependence shows up in δα. In order to remove such gauge dependence, one can add pinch-term contributions to the above 2-point functions. Pinch-terms are the "propagator-like" part of vertex and boxdiagram corrections to a 2-to-2 fermion scattering process (i.e., ff → ff with f and f being SM fermions), and can be extracted by cancelling internal fermion propagators using a contracted loop momentum in the numerator. Here the extracted pinch-terms do not depend on the choice of the external fermions. Since the pinch-terms also depend on the gauge choice, they can be expressed in a way similar to Eq. (47) as where One can verify that the second term of Eq. (50) satisfies the property Π PT . Therefore, the 2-point functions with the pinch-terms Π φ 1 φ 2 ≡ Π φ 1 φ 2 + Π PT φ 1 φ 2 are gauge-independent, and one should consider the gauge-independent counterterm δᾱ defined by instead of δα, as we will do in the following discussions.

B. CP-odd part
Next, we discuss the gauge dependence of the 2-point functions for the CP-odd scalar The gauge-dependent part is expressed as Analogous to the CP-even case, we can add the pinch-terms extracted from a ff → ff process: where the gauge-independent part while the gauge-dependent part Analogous to Eq. (52), one should consider the pinch term-included counterterm δβ defined by where in this case; that is, δβ still has explicit gauge dependence. In fact, the 5-plet Higgs boson loop contributions to Π A 1 A 2 G.D. , the terms proportional to ζ A 1 ζ A 2 and c H 0 5 A 1 Z c H 0 5 A 2 Z in Eq. (54), are not cancelled by the pinch-terms because of the fermio-phobic nature of the 5-plet Higgs bosons. Therefore, even after the pinch-terms are included, gauge dependence still remains  [15,35].
To see the ξ dependence, we introduce where δβ fin is the finite part of δβ. In Fig. 1, we show how the renormalized mixing anglê β depends on the choice of the gauge parameter ξ (= ξ W = ξ Z ). We see that the gauge dependence ofβ becomes larger for larger values of ξ and/or v ∆ , but it is at most about 1% or smaller when ξ ≤ 10 3 . Therefore, in the numerical evaluation for the renormalized Higgs boson vertices, the actual effect from the gauge dependence in δβ is negligibly small. Moreover, the modifications cause by varying α between 0 • and −20 • , a range of phenomenological interest, are much minor. In the following discussion, we will use δβ instead of δβ.
Before closing this section, we would like to remark that the gauge dependence of the 2-point function for the CP-odd Higgs bosons is cancelled if we add up all the contributions to the ff → ff scattering amplitude from the H 0 3 -H 0 3 , H 0 3 -G 0 and G 0 -G 0 mediators: One can explicitly verify that the gauge dependence in the above expression is exactly cancelled among the three terms in the square brackets.

V. RENORMALIZED HIGGS VERTICES
In this section, we compute renormalized hV µ V ν (V = W, Z), hff and hhh vertices based on the on-shell scheme discussed in Section III. We note that the on-shell conditions are insufficient to fix all the counterterms appearing in the renormalized hhh vertex. Therefore, we have to introduce an additional condition, the minimal subtraction (MS) scheme, to be discussed in Section V C. All the analytic expressions for the 1PI diagram contributions (variables labeled with the superscript "1PI") to 1-, 2-and 3-point functions are given in Appendix C.
Hereafter, we use the shorthand notation for the trigonometric function as s θ = sin θ, c θ = cos θ and t θ = tan θ.

A. Renormalized hV V vertex
The renormalized hV µ V ν vertices can be generally decomposed as: where p µ 1,2 and q µ (= p µ 1 +p µ 2 ) are the incoming momenta of the gauge bosons and the outgoing momentum of h, respectively. Each of the renormalized form factorsΓ i hV V can be further decomposed into four parts as: where Γ i,tree hV V , δΓ i hV V , Γ i,1PI hV V and T i hV V denote the contributions from tree-level diagrams, counterterms, 1PI diagrams and tadpoles, respectively. We note that the tadpole part Π Tad AB in the counterterms is here grouped into T i hV V . According to Eq. (26), Each term in Eq. (62) is given as follows: As mentioned in the previous section, we adopt the pinched counterterm δβ defined in Eq. (58) instead of δβ given in Eq. (45).

B. Renormalized hff vertex
The renormalized hff vertices can be expressed in terms of eight form factors as follows: where p µ 1,2 and q µ (= p µ 1 + p µ 2 ) are the incoming momenta of the fermions and the outgoing momentum of h, respectively. Analogous to the renormalized hV µ V ν vertices, each of the renormalized form factors is further decomposed into the following four parts: where i = S, P, V 1 , V 2 , A 1 , A 2 , T, P T . We note that the tadpole term cannot be inserted to the tree-level hff diagram; that is, Γ Tad hf f = 0. Hence, the tadpole contribution T i hf f is obtained only from the corresponding counterterm. Each term of Eq. (66), except for the 1PI part, is given as follows: with Γ i,tree hf f = δΓ i hf f = T i hf f = 0 for i = S. As for the renormalized hV µ V ν vertex, we also use the pinched counterterm δβ in the contribution to the hff vertex.

C. Renormalized hhh vertex
Finally, we compute the renormalized hhh vertex which is trivial in the Lorentz structure as it is a scalar vertex. Analogous to the hV µ V ν and hff vertices, the renormalized hhh vertex can be expressed aŝ where p µ 1,2 and q µ (= p µ 1 + p µ 2 ) are the incoming and outgoing momenta for the Higgs boson, respectively. Each of the contributions is given as follows: where λ φ i φ j φ k and λ φ i φ j φ k φ l are defined in Eq. (C9), and The counterterm δλ hhh is expressed as Notice here that δα and δβ are correctly replaced by the corresponding pinched counterterms δᾱ and δβ.
In Eq. (71), the counterterms δµ 1 and δµ 2 show up and cannot be individually determined by applying the on-shell scheme, as alluded to in the beginning of the section. A similar situation also happens in THDMs [8] and the HSM [6]. In this paper, we apply the MS scheme to fix the combination of δµ 1 and δµ 2 , where these counterterms are determined so as to cancel only the UV divergent part ∆ div of δΓ hhh (without the δµ 1 and δµ 2 terms), Γ 1PI hhh and T hhh . Here, ∆ div ≡ 1/ + ln 4π − γ E + ln µ 2 with µ and being defined in Appendix C 1 and γ E being the Euler-Mascheroni constant. The same method has also been applied to fix the counterterm for the hhh vertex in THDMs [8,11] and that in the HSM [6].

D. Renormalized Higgs boson couplings
We can now calculate the renormalized Higgs boson vertices (i.e., hV V , hff and hhh vertices) from the discussions in the previous subsections. We here define the renormalized scale factorsκ X for the Higgs boson couplings, which are convenient to discuss the deviation in the couplings from the SM predictions, as follows: where p 2 denotes the squared momentum for the off-shell particle, namely, V * , t * and h * for the hV V , htt and hhh couplings, respectively. For the hbb and hτ + τ − couplings, we define their renormalized scale factors without the momentum dependence since the on-shell decays h → bb and h → τ + τ − are allowed: For the numerical evaluation of these scale factors, we use the following SM input parameters [21]: We only show the points allowed by the constraints of perturbative unitarity and vacuum stability,   and notice δZ h = −Π hh (m 2 h )) defined in Eq. (C10), which can be significant depending on µ 1 and µ 2 , and it determines the typical size of quantum corrections.
Similar toκ W andκ Z , the behavior ofκ t is roughly determined by the tree-level prediction, i.e., c hf f given in Eq. (B8). In fact, it is seen thatκ t becomes small when we take larger values of v ∆ and |α|. In addition, the quantum correction reducesκ t , mainly because of the effect of δZ h . We note that the predictions forκ b andκ τ are almost the same as that ofκ t .
Forκ h , there are several features different fromκ W ,κ Z andκ t . First of all, the tree-level prediction, shown by the green dots, is not a single-valued curve, but spreads over a region on the plane. This is becauseκ h depends not only on α and v ∆ but also on µ 1 and µ 2 as seen in Eq. (69). Secondly,κ h can receive a large quantum correction at several 100% level with respect to the tree-level prediction. This large correction can be ascribed to the λ 3 While the predicted scale factors presented here are only for some special cases, we will show their behaviors in more generic cases in Section VII.
Finally, we discuss the momentum dependence ofκ W ,κ Z ,κ t andκ h . We provide six benchmark points (BP1-BP6), all of which are allowed by both the constraints of perturbative unitarity and vacuum stability. In Table I In Fig. 3, the momentum dependence ofκ V ,κ t andκ h are shown for the six benchmark points. Forκ V , bothκ W andκ Z monotonically increase with p 2 , where the increasing rates for BP2, BP4 and BP6 are more significant as compared to those for BP1, BP3 and BP5.
We also observe that the increasing rates becomes slightly higher at p 2 800 GeV because of the threshold effects of the extra Higgs bosons. In addition, the difference betweenκ W andκ Z is getting larger as p 2 increases. On the other hand, the momentum dependence

VI. CONSTRAINTS ON PARAMETER SPACE
In this section, we discuss both theoretical and experimental constraints that we impose on the model parameters. A search of viable exotic Higgs boson mass spectra based upon similar constraints and prospects for detecting the doubly-charged Higgs boson at the 14-TeV LHC and a 100-TeV future pp collider had been studied in Ref. [31].

A. Theoretical bounds
Two theoretical constraints are taken into account to constrain the dimensionless quartic couplings of the scalar potential at tree level: the stability of the electroweak vacuum and the unitarity of the perturbation theory. These constraints on the quartic couplings can be translated into bounds on the physical parameters such as the masses and mixing angles of the Higgs bosons through the relations given in Appendix A.
The vacuum stability requires the scalar potential to be bounded from below and leads to the following constraints for the quartic couplings [32]: where with B randomly varying between 0 and 1.
The bound from perturbative unitarity is obtained by requiring that the s-wave amplitude matrix, a 0 , for elastic 2 → 2 scalar boson scatterings does not become too large to violate S-matrix unitarity. One can set the criteria for this requirement as that the magnitudes of all the eigenvalues of a 0 do not exceed 1/2. In the high-energy limit, the matrix elements of a 0 are expressed by the scalar quartic couplings because only the diagrams involving scalar contact interactions are relevant. In this setup, one can obtain the following conditions [32,41]: We note that by combining the vacuum stability condition, the first 2 inequalities of (77) can be simply replaced by (78)

B. Experimental bounds
Next, we discuss the experimental constraints from the electroweak oblique S parameter, the signal strengths for the 125 GeV Higgs boson, and the direct searches for extra Higgs bosons. We note that the oblique T parameter is used as one of the inputs (see the discussion in section III A), so that it cannot be applied to the constrain the GM model. We require that predictions of these observables in the model be within the 95% confidence level (CL) region.
In the following, we explain how these constraints from experimental data are imposed in our analysis, in order.
The Higgs signal strengths have been measured from 20 channels with different combinations of production and decay channels in Ref. [42]. Among these measurements, we do not include the signal strengths for the Zh and th productions with the h → W W * decay in our study because the SM predictions for these two channels are excluded by the current data at 95% CL. It should be noted that channels with h decaying into a pair of photons provide effective constraints on the masses of extra Higgs bosons as their dependences appear in the charged Higgs boson (H ± 3 , H ± 5 and H ±± 5 ) loop contributions to the h → γγ decay. In contrast, all the other channels depend on only two parameters: α and β.

C. Allowed parameter space
We are now ready to present the allowed parameter space by imposing the constraints discussed in the previous subsections.
After fixing v and m h , there are totally seven independent free parameters in the GM model as shown in Eq. (11), assuming the custodial symmetry at tree level. Instead of using the parameters given in Eq. (11), we choose four dimensionless quartic couplings λ 2−5 and three dimensionful parameters µ 1,2 and m ∆ in the Higgs potential as our inputs, with which all the other parameters are determined. We then perform a scan of the parameters in the following ranges: The ranges of λ 2,3 are determined by the constraints from perturbative unitarity and vacuum stability, while those of λ 4,5 are determined by the bounds from the perturbative unitarity only [32]. The parameter scan is performed under two sets of constraints: Set-A takes into account the constraints of vacuum stability, perturbative unitarity and the S parameter, and Set-B further considers the Higgs signal strengths and direct search of H ±± 5 , all at 95% CL.
In Fig. 4, points allowed by Set-A constraints (left plot) and Set-B constraints (right plot) with the special case in Fig. 3. We note that within our parameter scan ranges,κ Z varies from 0.88 (0.93) to 1.12 (1.13) for p 2 = 250 (500) GeV. On the other hand, from the right plots we see that maximal |∆κ V | is typically around 0.2% for p 2 = 250 GeV, while the maximum becomes around 0.8% for p 2 = 500 GeV. It is also seen that |∆κ V | does not depend on v ∆ and α so much. We note that ∆κ V can be either positive or negative, and it falls in the range of −0.7% to 0.4% for p 2 = 250 GeV and 0.0% to 1.45% for p 2 = 500 GeV.
In Fig. 6, we show the scatter plots ofκ b (left) andκ t with p 2 = 500 GeV (right) in the α − v ∆ plane. As shown, the behaviors ofκ b andκ t are almost the same as each other.
The result forκ τ is also very similar to that ofκ b . In contrast to the case ofκ Z , the value ofκ b becomes smaller when |α| becomes larger. Finally, we show the correlation of the renormalized scale factors. Fig. 8 shows the correlation betweenκ Z andκ τ , where the momentum p 2 ofκ Z is set to be 250 GeV and 500 GeV in the left and right plots, respectively. We see that the distribution of the dots in theκ Z -κ τ plane for p 2 = 500 GeV is almost the same as that for p 2 = 250 GeV, except for slight shrinking in the range ofκ Z in the former case. It is also seen that the range of possible κ τ gets restricted whenκ Z becomes larger. Atκ Z 1.13,κ τ is predicted to be about 0.95. Fig. 9 shows the correlation betweenκ Z andκ h , where the momentum p 2 ofκ Z is set to be 250 GeV and 500 GeV in the left and right plots, respectively, while that ofκ h is fixed at 500 GeV for both plots. Aside from some shifting in the dot distributions between the two plots, most of the predictedκ h values are between 1 and 5. Again, the possible range of κ h is restricted whenκ Z becomes larger. In particular,κ h is predicted to be about 1 when κ Z 1.13. We also notice some of the predictedκ h values are less than 1 or even negative In this appendix, we give the mass eigenstates of the scalar fields in the GM model and their masses as derived from the potential in Eqs. (6) and (7).
The mass eigenstates of the scalar fields are related to the original fields given in Eq. (1) by the following transformations: where G ± and G 0 are the NG bosons to become the longitudinal components of W ± and Z bosons, respectively. The rotation matrices in Eq. (A1) with the mixing angles satisfying The other mixing angles γ, α 1 , α 2 and α 3 generally have very complicated forms. Nevertheless, an important thing is that in the ν → 0 limit γ, α 1 and α 2 become zero, while α 3 can be nonzero.
The squared masses of the physical Higgs bosons are given by where and It is observed that in the ν → 0 limit, the different charged states within each multiplet have the same mass as the consequence of the restoration of the custodial symmetry.

Appendix B: Interaction terms of the Higgs bosons
We give expressions for the relevant 3-point and 4-point interaction terms of the Higgs bosons. The scalar-gauge-gauge interaction terms are given by where and The scalar-scalar-gauge interaction terms are given by The scalar-scalar-gauge-gauge interaction terms are given by where The Yukawa interaction terms for the third-generation fermions are given by where I t = 1/2 and I b = I τ = −1/2, and The B and C tensor functions are decomposed into the following forms in terms of scalar coefficients B 1,21,22 and C 11,12,21,22,23,24 : It is convenient to define coefficients λ φ i φ j φ k and λ φ i φ j φ k φ l respectively for the 3-point and 4-point scalar interaction terms as As some of these coefficients are proportional to each other, we thus define the following quantities: and with φ = h or H 1 .