A scale-invariant Higgs sector and structure of the vacuum

Abstract

Abstract

In view of the current status of measured Higgs boson properties, we consider a question whether only the Higgs self-interactions can deviate significantly from the Standard-Model (SM) predictions. This may be possible if the Higgs effective potential is irregular at the origin. As an example we investigate an extended Higgs sector with singlet scalar(s) and classical scale invariance. We develop a perturbative formulation necessary to analyze this model in detail. The behavior of a phenomenologically valid potential in the perturbative regime is studied around the electroweak scale. We reproduce known results: the Higgs self-interactions are substantially stronger than the SM predictions, while the Higgs interactions with other SM particles are barely changed. We further predict that the interactions of singlet scalar(s), which is a few to several times heavier than the Higgs boson, tend to be fairly strong. If probed, these features will provide vivid clues to the structure of the vacuum. We also examine Veltman's condition for the Higgs boson mass.

Published for SISSA by Springer

Received: March 18, 2015 Accepted: April 2, 2015

Published: May 6, 2015

A scale-invariant Higgs sector and structure of the vacuum

K. Endo and Y. SuminoDepartment of Physics, Tohoku University,

Sendai, 980-8578 Japan

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: In view of the current status of measured Higgs boson properties, we consider a question whether only the Higgs self-interactions can deviate signicantly from the Standard-Model (SM) predictions. This may be possible if the Higgs e ective potential is irregular at the origin. As an example we investigate an extended Higgs sector with singlet scalar(s) and classical scale invariance. We develop a perturbative formulation necessary to analyze this model in detail. The behavior of a phenomenologically valid potential in the perturbative regime is studied around the electroweak scale. We reproduce known results: the Higgs self-interactions are substantially stronger than the SM predictions, while the Higgs interactions with other SM particles are barely changed. We further predict that the interactions of singlet scalar(s), which is a few to several times heavier than the Higgs boson, tend to be fairly strong. If probed, these features will provide vivid clues to the structure of the vacuum. We also examine Veltmans condition for the Higgs boson mass.

Keywords: Higgs Physics, Spontaneous Symmetry Breaking, Beyond Standard Model

ArXiv ePrint: 1503.02819

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP05(2015)030

Web End =10.1007/JHEP05(2015)030

JHEP05(2015)030

Contents

1 Introduction 1

2 Model and e ective Higgs potential 32.1 Lagrangian 32.2 E ective potential up to one-loop level 42.3 Renormalization group analysis 5

3 Perturbatively valid parameter region 63.1 Order counting in perturbative expansion 63.2 Comment on the Hessian matrix 93.3 Relation between H and HS at [notdef] = vH 10

4 Results of analysis 104.1 Phenomenologically valid parameters 104.2 Interactions among physical scalar particles 164.2.1 Our results 164.2.2 Comparison with results using Gildener-Weinbergs framework 174.3 Veltmans condition for the Higgs mass 19

5 Conclusions and discussion 21

1 Introduction

It appears that completion of the standard model (SM) of particle physics, as it stands, has been achieved with the discovery of the Higgs boson at the LHC experiments [1, 2]. In particular, the two parameters of the Higgs potential,

VSM(H) = [notdef]H2HH + H [parenleftBig]

2, (1.1)

namely, the vacuum expectation value (VEV) of the Higgs eld vH = [notdef]H/p H = 246 GeV

and the Higgs boson mass mh = p2[notdef]H 126 GeV, are now determined. Here, H denotes

the Higgs doublet eld H = (H+, H0)T. This means that all the parameters of the SM have been determined. After the discovery, investigations of properties of the Higgs boson are being performed rapidly, such as measurements of its spin, CP , and couplings with various SM particles. Up to now, there are no evident contradictions with the SM predictions. Accuracies of the measurements are improving, and identication with the SM Higgs boson is becoming more likely. These features, however, do not readily lead to the conclusion that the SM is conrmed altogether. As an important piece, conrmation

1

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HH

of the self-interactions of the Higgs boson is still missing, which is indispensable to unveil the structure of the Higgs potential.

There are some phenomenological and theoretical problems in the Higgs sector of the SM as of today. For instance, there is a huge hierarchy between the electroweak and Planck scales, which generates the Naturalness problem. Furthermore, even if we permit ne tuning of parameters, we face a problem that the vacuum of the Higgs potential is unstable or metastable at a high-energy scale.1 These problems give us motivations to consider extensions of the SM Higgs sector.

In view of the present status, it is an interesting question whether it is possible, with such an extension, that only the Higgs self-interactions are signicantly di erent from the SM predictions, while other properties of the Higgs boson are barely a ected. Naively, one may expect that deviations of the Higgs self-couplings from the SM values can be expressed as e ects of higher-dimensional operators, which are suppressed by a cut-o scale as (vH/ )n. It is based on a general model-independent argument in the case that the Higgs e ective potential can be expanded as a polynomial in the Higgs eld H. Since the cut-o scale may be of order a few TeV scale or higher, one may expect that the deviation is already constrained to be quite small.

One direction to evade such an argument is to consider models with non-decoupling e ects to Higgs self-interactions. There are many examples [3, 4]. Here, we would like to consider a di erent possibility. In various physics of spontaneous symmetry breakdown (including condensed matter physics), there appear e ective potentials which cannot be expanded in polynomials in the eld variables, namely e ective potentials which are irregular at the origin. For example, the theory of superconductivity at T = 0 gives such an e ective potential, and in certain strongly interacting systems such singular potentials are also expected to appear (although they are di cult to compute reliably). As an example which is perturbatively computable within relativistic quantum eld theory, we consider an extended Higgs sector with classical scale invariance. In this case, typically the Higgs e ective potential takes a form 4

, which is irregular at the origin via the Coleman-Weinberg (CW) mechanism [5], where = p2 Re H0. If this potential is expanded about the VEV in terms of the physical Higgs eld h = vH, the expansion

generates powers of h higher than the quartic power, and they arise in powers of h/vH without being suppressed by a cut-o scale. At the same time, it is expected that the triple and quartic couplings of h can have order-unity deviations from the SM values.

In recent years, CW-type potentials have been studied extensively as models of an extended Higgs sector, with di erent motivations. For instance, classical scale invariance as a solution to the hierarchy problem has become a hot subject. These are models which become scaleless at a new physics scale M (e.g. Planck scale), [notdef]H2(M) = 0. In these

models, typically the scale invariance is broken radiatively by the CW mechanism, which (in steps) leads to a generation of the electroweak scale as the Higgs VEV [613], or generates scales involving nonzero VEVs of other scalar elds [1428]. Even without explicit

1The scale of the vacuum instability is very sensitive to the top quark mass and depends on our precise knowledge of the top quark mass in the future.

2

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ln

2/vH2[parenrightbig]

1/2

realizations, classical scale invariance is often mentioned as a possibility or guideline for an underlying mechanism.

The purpose of this paper is to explore physics in the vicinity of our vacuum, characterized by the e ective potential of an extended Higgs sector. As a minimal extension of the SM Higgs sector, we consider a scale-invariant Higgs potential with a real singlet scalar eld which belongs to the N representation of a global O(N) symmetry. Though there have been several works in which the same or similar extensions are considered, we re-analyze the model by examining the validity range of perturbative calculations of the e ective potential in detail. We can obtain a phenomenologically valid potential within the validity range of a perturbative analysis. We clarify its behavior around the electroweak scale. We also examine Veltmans condition [29] for the Higgs mass, which is a criterion for judging a Naturality of the scalar potential by examining the coe cient of the quadratic divergence.

Some of previous works on similar models are subject to possible instability of their predictions when higher-order perturbative corrections are included. Such instability was already pointed out by the original CW paper, and we re-examine this condition. Other works, which are legitimate with respect to the perturbative validity, use the framework of Gildener-Weinberg (GW) [30] to analyze the e ective potential. In this framework, one concentrates on a one-dimensional subspace of the conguration space of the e ective potential, namely that corresponding to the physical Higgs direction, and examines the potential shape on that subspace. On the other hand, we analyze the e ective potential in a way closer to the original CW approach, which enables to clarify a global structure of the potential shape in the conguration space.

Ref. [9] analyzed a gauged and non-gauged version of a scale-invariant model, and the latter is close to the one we examine. Phenomenologically we reproduce similar aspects, hence we state the di erences of our analysis in comparison. As mentioned above, ref. [9] uses the GW framework to analyze the properties of the e ective potential. Instead we present a formulation which enables analysis of global properties of the e ective potential. As a result, for instance, we are able to analyze the structure of the potential in every direction around the vacuum in the conguration space and predict the interactions among the physical scalar particles.

This paper is organized as follows. In section 2 we explain our model. In section 3, we develop a theoretical framework needed for a perturbative analysis of the model. We give results of our numerical analysis in section 4. Conclusions and discussion are given in section 5.

2 Model and e ective Higgs potential

2.1 Lagrangian

The scale-invariant limit of the SM has long been excluded experimentally. Hence, to impose classical scale invariance, we need to extend the Higgs sector. As a minimal extension, we consider a scale-invariant extension of the SM with an additional real singlet scalar eld with a Higgs-portal coupling. We consider the case where the singlet scalar eld is in the

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fundamental representation of a global O(N) group: = (S1, [notdef] [notdef] [notdef] , SN)T. The Lagrangian is given by

L = [LSM][notdef]H!0 +

1

2(@[notdef]

)2 , (2.1)

where the real singlet eld interacts with itself and the Higgs doublet eld H via the self-interaction and portal interaction with the coupling constants S and HS, respectively.

2.2 E ective potential up to one-loop level

The one-loop e ective potential in the Landau gauge, renormalized in the MS-scheme, is given by

Ve (, ') = Vtree(, ') + V1-loop(, ') , (2.2)

Vtree(, ') = H4 4 +

)2 HS(HH)(

[notdef]

)

S

4 (

[notdef]

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2 2'2 +

4 '2 , (2.3)

V1-loop(, ') =

HS

S

Xi

ni4(4)2 Mi4(, ') [bracketleftbigg]ln Mi2(, ')

[notdef]2 ci

[bracketrightbigg]

. (2.4)

Here, the expectation values of the scalar elds in the presence of source J are given by

hH[angbracketright]J =

1 p2[parenleftBigg][parenleftBigg]

0

[parenrightBigg]

, [angbracketleft]

[angbracketright]J = (', 0, [notdef] [notdef] [notdef] , 0)T ; , ' 2 R . (2.5)

The index i denotes the internal particle in the loop, and their parameters are given by2

W bosons: nW = 6 , MW 2 = 14g22 , cW =

5

6 ;

Z boson: nZ = 3 , MZ2 = 14(g2 + g[prime]2)2 , cZ =

5

6 ;

massive scalar bosons: n[notdef] = 1 , M[notdef]2 = F[notdef] , c[notdef] =

3 2 ;

NG bosons of the Higgs eld: nNG = 3 , MNG2 = H2 + HS'2 , cNG = 32 ;

NG bosons of the singlet eld: nNG = N 1 , MNG2 = HS2 + S'2 , cNG =

3 2 ;

up-type quarks (F = u, c, t): nF = 4NC , MF 2 =

1

2

y(U)FF

22 , cF = 32 ;

down-type quarks (f = d, s, b): nf = 4NC , Mf2 =

1

2

y(D)ff

22 , cf = 32 ;

22 , cf = 32, (2.6)

2Strictly speaking, the terminology Nambu-Goldstone (NG) bosons for the internal particles is inadequate except at the vacuum conguration and depends on how the symmetries are broken by the vacuum. More precisely, NG bosons represent the scalar modes which are orthogonal to the radial directions of H and S.

charged leptons (f = e, [notdef], ): nf = 4 , Mf2 =

1

2

y(E)ff

4

and F[notdef] are dened by

F[notdef](, ') =

2 2 +

2 '2

3 H + HS

HS + 3 S

[notdef]

[radicalBigg][parenleftbigg]

3 H HS

2 2 +

HS 3 S

2 '2

2+ 4 HS22'2 . (2.7)

Hereafter, we neglect all the Yukawa couplings except the top Yukawa coupling, yt =

y(U)33 O(1), since other Yukawa couplings are fairly small. It is customary to denote the

summation over i together with particles statistical factor as supertrace STr, which we also use below.

2.3 Renormalization group analysis

We can extend the applicability range of the e ective potential by a renormalization-group (RG) improvement. According to the general formulation [31, 32], the (L + 1)-loop beta functions and anomalous dimensions can be used to improve the L-loop e ective potential. Here, we obtain the leading-logarithmic (LL) potential by improving the tree-level potential by the one-loop beta functions and anomalous dimensions. It is argued in [31, 32] that, within the range where ln(Mi2/[notdef]2) are not too large, combining the one-loop e ective potential with the one-loop beta functions and anomalous dimensions gives a better approximation. Hence, we obtain an improved-next-to-leading-order (improved-NLO) potential by combining them. Comparing NLO, LL and improved-NLO potentials, we can examine validity (stability) of the predictions in the vicinity of the vacuum.

The beta functions and anomalous dimensions are dened by

X = [notdef]dX

d[notdef] , A =

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dA

d[notdef] , (2.8)

respectively, where X is a coupling constant and A is a eld, both renormalized at scale [notdef]. The one-loop beta functions and anomalous dimensions of the model are given by

H = 1

(4)2

98g4 +34g2g[prime]2 +38g[prime]4 6yt4

4

94g2 +34g[prime]2 3yt2[parenrightbigg]

H + 24 H2 + 2N HS2[bracketrightbigg]

, (2.9)

HS = 1

[notdef] A

(4)2 HS

2

94g2 +34g[prime]2 3yt2[parenrightbigg]+ 12 H + 8 HS + 2(N + 2) S

[bracketrightbigg]

, (2.10)

S = 1

(4)2 8 HS2 + 2(N + 8) S2

[parenrightbig]

, (2.11)

= 1

(4)2

94g2 +34g[prime]2 3yt2[parenrightbigg]

, (2.12)

' = 0 . (2.13)

The beta functions which do not include HS or S are suppressed.

5

We obtain the improved potentials by taking the renormalization scale as t = ln(

p2 + '2/vH).

V (LL)e (, ') =

H(t)

4 4(t) +

HS(t)

2 2(t)'2(t) +

S(t)

4 '4(t) , (2.14)

V (imp-NLO)e (, ') =

H(t)

4 4(t) +

HS(t)

2 2(t)'2(t) +

S(t)

4 '4(t)

+

Xi

ni4(4)2 Mi4 ((t), '(t)) [bracketleftbigg]ln Mi2 ((t), '(t)) [notdef]2(t) ci

[bracketrightbigg]

, (2.15)

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where

[notdef](t) = vH et , (2.16) (t) = (t) , (2.17)

'(t) = '(t)' , (2.18)

i(t) = exp

[bracketleftbigg][integraldisplay]

t

0 i(t[prime])dt[prime][bracketrightbigg]

. (2.19)

3 Perturbatively valid parameter region

In a previous work [16] a parameter region has been searched assuming

H HS < 1 . (3.1)

In that case, however, the Coleman-Weinberg mechanism does not work properly, since the one-loop corrections cannot compete against the tree-level terms in such a parameter region if the order counting in perturbation theory is legitimate. (This is in analogy to the case of pure scalar 4 eld theory considered in the original CW paper [5].) As a result, the renormalization scale considered in [16] is about 105vH, where the large logarithmic corrections invalidate a perturbative analysis. In this section, we reconsider the parameter region, in which the results are perturbatively valid and the CW mechanism works properly.

As mentioned in the Introduction, there is a framework of analysis to nd a vacuum of a one-loop e ective potential and to compute particle contents of scalar bosons systematically, which was introduced by Gildener and Weinberg [30]. In that framework we obtain the following general results. A perturbatively valid vacuum appears on the ray of a at direction of the tree-level potential; a light scalar boson scalon (corresponding to the Higgs boson in our context) appears in addition to heavy scalar bosons; interactions among the scalon and the other scalar bosons are derived. Although it is a useful framework, we consider that our framework presented below is advantageous to analyze more global features of the potential, including the vicinity of the vacuum.

3.1 Order counting in perturbative expansion

We investigate an order counting among the coupling constants, with which the CW mechanism is expected to work in the perturbative regime. To realize the CW mechanism, the

6

tree-level and one-loop contributions to the e ective potential should be comparable and compete with each other. Parametrically this requires a relation

| H[notdef]

N HS2

NC yt4

(4)2 1 (3.2)

to be satised. (This is in analogy to the case of massless scalar QED considered in the original CW paper [5], in which

e4 (4)2

(4)2

1 is required as a consistent parameter region.)

Note that we take yt into account in eq. (3.2) since the top quark gives the dominant one-loop contribution among the SM particles which couple to the Higgs particle. According to eq. (3.2), we consider that H and HS2 (as well as yt4) are naively counted as the same order quantities in perturbative expansions.3 It follows that the relation between H and HS should read

| H[notdef] [notdef] HS[notdef] . (3.3)

Thus, eq. (3.2) and eq. (3.3) indicate that H2 and HS2 need to be counted as di erent orders, although they both belong to the one-loop contributions. Furthermore, HS needs to be large, at least of order

pNC/N yt 1.7 N1/2, in order to beat the top quark negative contribution, for stabilizing the vacuum.

We derive a systematic approximation of the e ective potential eq. (2.4), taking into account the above order counting. First, the contributions from the NG bosons of the Higgs eld on the right-hand-side of eq. (2.4) can be written as follows:

3642 ( H2 + HS'2)2 [bracketleftbigg]

ln

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H2 + HS'2[notdef]2[parenrightbigg] 32[bracketrightbigg]

HS'2[notdef]2[parenrightbigg] 32[bracketrightbigg]

. (3.4)

Secondly, F[notdef](, ') in the fourth and fth terms in eq. (2.4) can be written as follows:

F[notdef](, ') [similarequal]

[similarequal]

3642 HS'2

2 ln

2 2 +

2 '2

HS

HS + 3 S

HS

2+ 4 HS22'2

F[notdef]app(, ') , (3.5)

where we have dened F[notdef]app(, ') as the approximate form of F[notdef](, '). Then we substi

tute eqs. (3.4) and (3.5) to the expression of the e ective potential eq. (2.4).

We also have to apply eq. (3.2) or eq. (3.3) to the tree-level potential in eq. (2.3). In particular, eq. (3.2) can be interpreted as follows. Although the term proportional to H is tree-level, after taking into account the above order counting of the coupling constants, this term should be regarded as next-to-leading order (NLO), in contrast to the term

3We do not consider a ne cancellation between the HS2 and yt4 contributions.

7

[notdef]

[radicalBigg][bracketleftbigg]

2 2 +

HS 3 S

2 '2

Figure 1. The LO e ective potential VLO as a function of and '. The red line shows the axis, which composes the minima of the potential.

proportional to HS, which is at the LO. For this reason, the LO contributions to the e ective potential in the above order counting is given by

VLO = HS2 2'2 +

4 '4. (3.6)

We show VLO in gure 1. At LO, the potential is at along the axis, which composes the minima of this potential.4 Thus, at LO, the vacuum is not determined uniquely. At every vacuum the Higgs boson is massless, while the singlet scalers are massive.

The NLO contributions are given by

VNLO = H4 4 +

F+app2(, ')

642

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S

ln

F+app(, ')

[notdef]2[parenrightbigg] 32[bracketrightbigg]

+ 3

642 HS'2

2 ln

HS'2[notdef]2[parenrightbigg] 32[bracketrightbigg]

+ N 1

642 HS2 + S'2

2 ln

HS2 + S'2[notdef]2[parenrightbigg] 32[bracketrightbigg]

4NC642 Mt4() [bracketleftbigg]

ln

Mt2()

[notdef]2[parenrightbigg] 32[bracketrightbigg]

+ 6

642 MW 4() [bracketleftbigg]

ln

MW 2()

[notdef]2[parenrightbigg] 56[bracketrightbigg]

+ 3

MZ2()

[notdef]2[parenrightbigg] 56[bracketrightbigg]

. (3.7)

We explain the reason for omitting Fapp in the next subsection.

As seen above, at every vacuum of VLO, ' = 0 and the '-direction becomes a massive mode. Hence, from consistency of the perturbative expansion, we expect that ' = 0 holds also at the vacuum of VLO + VNLO. According to the general argument on the potential

4If S is small and of the same order as H, S'4 should be counted as NLO. In this case, both and ' axes become the at minima of VLO. Other features, especially the results presented in the next section, are hardly a ected, if the global minimum of VLO + VNLO is on the axis.

8

642 MZ4() [bracketleftbigg]

ln

with a at direction at LO, with an appropriate choice of the couplings ( H, HS, S),

VLO +VNLO exhibits a minimum on the -axis by the CW mechanism, and the Higgs boson becomes massive at NLO. Hence, the Higgs boson is generally expected to be much lighter than the singlet scalars. Thus, the SM gauge group is broken by the vacuum as required SU(2)L [notdef] U(1)Y ! U(1)EM, while the O(N) global symmetry is unbroken. [There appears

no NG mode with respect to the O(N) group.]

3.2 Comment on the Hessian matrix

A special prescription is needed for computing the CW potential, in the case that the minimum of the e ective potential is not determined uniquely at the LO of the perturbative expansion, such as in eq. (3.6).

Generally the arguments of logarithms in a one-loop e ective potential include the eigenvalues mi2 of the Hessian matrix of the tree-level scalar potential. The Hessian matrix is a matrix whose elements are given by @2Vtree@xi@xj , where xi denotes a scalar eld. Therefore,

the eigenvalues mi2 at the potential minimum coincide with the mass-squared eigenvalues of the scalar elds at tree level. In the case that the tree-level potential has a unique minimum, all the eigenvalues of the Hessian matrix at the minimum are positive, mi2 > 0,

and mi4 ln mi2 in the one-loop e ective potential are well-dened.5

Since in our case VLO is at along the -axis, m2 (the eigenvalue in the direction of ) can be negative at conguration points (, ') innitesimally away from the -axis. Instead, if we determine the potential minimum of VLO + VNLO and compute the Hessian matrix of VLO+VNLO in the vicinity of the minimum, all the eigenvalues are positive. Thus, a naive perturbative treatment is inappropriate in computing quantum uctuations in the vicinity of a vacuum in the case that there is a negative eigenvalue.

Here we adopt a prescription6 that (only) the eigenvalue m2 cannot be determined by VLO, and that this eigenvalue should be determined by VLO + VNLO, whose e ect through m4 ln m2 should be included in VNNLO.7 Since we do not compute VNNLO, in practice we

simply neglect its contribution. The other eigenvalue m'2 determined by VLO is positive. In eq. (2.6), the two eigenvalues are given by F[notdef](, '). The eigenvalues of the modes

orthogonal to these radial modes in the scalar sector are composed by those of the three degenerate modes of H and those of the N 1 degenerate modes of

, all of which are non-negative; see eqs. (2.6) and (3.4).8 We consider that these should be included in the computation of VNLO. Based on these considerations, we omit the contribution of Fapp

(the eigenvalue in the direction ) in eq. (3.7).

5For massless modes, such as NG bosons of the Higgs eld, we dene the values of mi4 ln mi2 in the limit mi2 ! 0, i.e., zero. See eq. (3.4).

6This corresponds to the following prescription for computing Ve by the background eld method. We determine the propagator of the relevant quantum eld not by the LO vertices alone but also by including one-loop self-energy corrections. Note that both LO vertices and one-loop self-energy corrections are dependent on the background elds (, ') and determined from VLO and VNLO, respectively.

7This is consistent in the vicinity of the -axis since m2 is an NLO quantity.

8At the vacuum, the three modes orthogonal to the radial mode of H are identied with the NG modes, whose eigenvalues vanish to all orders, while all the N modes of become degenerate, since the O(N) symmetry is unbroken.

9

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An analysis for a consistent treatment of the NG modes has been performed recently [33], which is similar in spirit to the above argument. We will further develop the above method for including higher-order corrections consistently in our future work.

3.3 Relation between H and HS at = vH

To study the e ective potential in the vicinity of the vacuum, a natural choice of the renormalization scale would be [notdef] = vH, where vH = 246 GeV is the VEV of the Higgs eld. Here, we set [notdef] = vH and ' = 0 and examine a relation among the scalar couplings ( H, HS, S). (Other couplings are xed to the SM values.) As a result we obtain a relation between H and HS.

Setting ' = 0, the e ective potential takes a form

Ve (, ' = 0) = C14 + C24 ln

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[notdef]

, (3.8)

where C1 and C2 are constants dependent on H, HS and independent of S. This shows that both mass and VEV of the Higgs boson are independent of S. [This is not the case if we use RG-improved potentials, since terms including both S and ln(

p2 + '2/vH) are

resummed.]

The potential minimum is determined by

@

@Ve (, ' = 0)

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

=vH

= 0, (3.9)

from which we obtain vH as a function of H, HS and [notdef]. Then we set [notdef] = vH:

vH = f( H, HS; [notdef]) [vextendsingle][vextendsingle][vextendsingle][vextendsingle]

[notdef]=vH

. (3.10)

Since vH is the only dimensionful parameter, the right-hand-side is proportional to vH and we can divide both sides by vH. This gives a relation between H and HS, which are renormalized at [notdef] = vH. The relation is shown in gure 2, obtained by solving eq. (3.10) numerically.9 Note that HS should be of order

pNC/N yt 1.7 N1/2 or larger (see section 3.1), while it should be smaller than order 4 to ensure perturbativity.

4 Results of analysis

4.1 Phenomenologically valid parameters

Using the e ective potential VLO + VNLO obtained in the previous section, we search for a phenomenologically valid parameter region for the couplings ( H, HS, S). We require

that the observed mass and VEV of the Higgs boson are reproduced.

9If we neglect the gauge couplings (and all the Yukawa couplings except yt), we obtain a simple relation:

H =

1162 [ yt4 (3 + 3 ln 2 6 ln yt) + N HS2(ln HS 1)]. (3.11)

This gives a good approximation of the numerical result.

10

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Figure 2. The relation between H and HS obtained at the vacuum (' = 0) with a choice of the renormalization scale [notdef] = vH, for various N. [Eq. (3.10)].

N 1 2 3 4 5 12 H(vH) 0.11 0.055 0.025 0.0045 0.012 0.075

HS(vH) 4.8 3.4 2.8 2.4 2.1 1.4

Table 1. Values of H and HS at [notdef] = vH xed by mh = 126 GeV and vH = 246 GeV, in the case Ve = VLO + VNLO.

The analysis in section 3.3 shows that, as long as we choose a renormalization scale [notdef] [similarequal] vH such that a perturbative analysis is valid close to the vacuum, the couplings H and

HS are related, as demonstrated in gure 2. The ratio of the Higgs boson mass and the VEV, mh/vH, varies along each line shown in the gure. Thus, if we x mh = 126 GeV and vH = 246 GeV, the values of H and HS are xed. The values of H and HS for various N are shown in table 1. These values are consistent with our order estimate eq. (3.2). If we use RG-improved potentials, H and HS are no longer xed, since they depend on the value of S.

Let us present results of our analysis for the case N = 1. Table 2 shows our predictions, together with representative input parameters. We compare the three di erent approximations of the e ective potential: (I) VLO + VNLO, (II) V (LL)e , and (III) V (imp-NLO)e .

Here, V (LL)e and V (imp-NLO)e are dened similarly to eqs. (2.14) and (2.15) from VLO and VLO + VNLO with the scale choice t = ln(

p2 + '2/vH), respectively. As mentioned, in the case (I), H and HS are xed by mh and vH, while S can be taken as a free input parameter. Reecting this feature, even in the cases (II) and (III), the values of H and HS are tightly constrained, while S can be taken fairly freely. For this reason, we take S as the input parameter for all three cases. In the table we take S = 0.10 as an example.

In all cases there exist H and HS which reproduce the mass and VEV of the Higgs boson. Furthermore, in accord with the argument in section 3.1, the mass of the singlet scalar is predicted to be several times larger than the Higgs boson mass in each case. We nd that

11

N = 1

(I) (II) (III) [notdef] vH = 246 [GeV]

yt(vH) 0.919

g(vH) 0.644

g[prime](vH) 0.359

H(vH) 0.11 0.059 0.082

HS(vH) 4.8 4.5 4.3

S(vH) 0.10 0.10 0.10

vH[GeV] 246 mh[GeV] 126

h'[angbracketright] [GeV] 0 0 0

ms[GeV] 556 527 524 sin mix 0 0 0

Landau pole [TeV] 3.5 4.1 4.7

Table 2. Predictions of our model, together with some representative input parameters, for N = 1 in three di erent approximations of the e ective potentials: (I) VLO + VNLO, (II) V (LL)e , and (III)

V (imp-NLO)e . In each case, a parameter set for ( H, HS, S) is chosen such that the mass and VEV of the Higgs boson are reproduced.

the predicted masses of the singlet scalar are consistent with each other within about 5% accuracy. The di erences may be taken as a reference for the stability of our predictions. An undesirable feature is that the locations of the Landau pole are close and in the several TeV region. This originates from the large value of HS at the electroweak scale [notdef] [similarequal] vH

(see section 3.1).

Dependences of the predictions on S is as follows. If we raise the value of S, the locations of the Landau pole are even lowered, since the couplings H, HS, S increase with the renormalization scale as they inuence each other. The mass of the singlet scalars are barely dependent on S, since only the fourth derivative in the ' direction of the e ective potential is a ected at LO. The values of H and HS do not change very much with S. For instance, if we take S = 0.5 and 1.0, the Landau pole appears at 3.2TeV and 2.8TeV respectively, while the mass of the singlet scalar changes little.

To see the shape of the e ective potential, we show in gure 3 (a) the contour plot of the potential in case (I) of table 2. Since the potential is symmetric under !

or ' ! ', we show only the upper-right part of the conguration space. As expected

there is a global minimum along the -axis. There is also a shallower local minimum along the '-axis, which is generated by the CW mechanism of a competition between S'4 and

HS2'4 ln( HS'2/[notdef]2) terms.10 In cases (II) and (III) of table 2, we obtain qualitatively similar contour plots.

10This local minimum on the '-axis is about ten times shallower than the global minimum. It becomes even shallower if the value of S is larger. We will not be concerned about the minimum on the '-axis in our analysis as long as it is not a global minimum.

12

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(a) (b)

Figure 3. (a) Contour plot of the unimproved e ective potential VLO + VNLO . (b) Comparison

of the e ective potentials on the -axis (' = 0), in the approximations (I) VLO + VNLO, (II) V (LL)e ,

and (III) V (imp-NLO)e . The e ective potential for the SM is also shown.

In gure 3 (b) we show the e ective potentials on the -axis (i.e., for ' = 0) for all three cases, together with the e ective potential of the SM. We see that di erences between the three approximations are small. In particular, the di erence between the LL approximation (II) and the improved-NLO approximation (III) is hardly visible. These features show good stability of our predictions in the vicinity of the vacuum, and that they are within the validity range of perturbation theory.

There is a signicant di erence between our e ective potential and that of the SM. In comparison to the SM, the minimum of our potential is shallow, although the values of vH and mh are common. The clear di erence of the potential shapes indicates that the higher derivatives of the potentials at the minimum are appreciably di erent. Namely, we anticipate that the Higgs self-interactions of our model are appreciably di erent from those of the SM.

Next we examine the cases N > 1. In general, we expect that the location of the Landau pole is raised as compared to the N = 1 case. This is because the required value of HS to overwhelm the top-loop contribution decreases with N as 1/pN, see section 3.1.

One may conrm this tendency in table 1. We show the cases N = 4 and N = 12 in table 3 with the input S = 0.10 at [notdef] = vH. As expected the positions of the Landau pole are raised up to order a few tens TeV for these N. The corresponding shapes of the e ective potentials are displayed in gures 4.

In gure 5 are shown the plots of the phenomenologically favored region for the portal coupling HS and the Higgs quartic coupling H. We show the results for the case (I), which are independent of S. The other parameters are xed to the SM values. These gures show how the parameters of the model under consideration are constrained in the current status. As we discussed, currently S is barely constrained.

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N = 4 N = 12(I) (II) (III) (I) (II) (III) H(vH) 0.0045 0.061 0.0005 0.075 0.063 0.082

HS(vH) 2.4 2.3 2.4 1.4 1.4 1.4

S(vH) 0.10 0.10 0.10 0.10 0.10 0.10

h'[angbracketright] [GeV] 0 0 0 0 0 0

ms[GeV] 378 378 375 285 293 286 sin mix 0 0 0 0 0 0

Landau pole [TeV] 16 19 17 28 37 26

Table 3. Predictions of our model for N = 4 and N = 12 in three di erent approximations of the e ective potentials: (I) VLO + VNLO, (II) V (LL)e , and (III) V (imp-NLO)e . In each case, a parameter set for ( H, HS, S) is chosen such that the mass and VEV of the Higgs boson are reproduced.

(a) (b)

(c) (d)

Figure 4. Same as gures 3 but for (a)(b) N = 4, and (c)(d) N = 12, corresponding to the parameters of table 3.

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(a) (b)

(c)

Figure 5. Plots for phenomenologically favoured region of the portal coupling HS and the Higgs quartic coupling H for (a) N = 1, (b) 4 and (c) 12, corresponding to the experimental data [34]. S is set as 0.1, although the results are fairly insensitive to S.

15

N = 1 N = 4 N = 12(I) (II) (III) (I) (II) (III) (I) (II) (III) hhh/ (SM)hhh 1.7 1.8 1.8 1.7 1.7 1.7 1.7 1.6 1.7

hhhh/ (SM)hhhh 3.7 4.3 4.5 3.7 3.2 3.4 3.7 2.8 3.1

hss 11.4 10.2 10.2 5.02 5.02 4.96 2.80 2.95 2.83

hhss 14 13 13 5.6 5.7 5.7 3.0 3.2 3.1

ssss 6.5 1.9 0.9

Table 4. Coupling constants among the scalar particles, corresponding to the parameters of tables 2 and 3. The coupling constants are dened in eq. (4.2).

4.2 Interactions among physical scalar particles

4.2.1 Our results

Let us expand the e ective potential about the vacuum after rewriting 2 ! HH, '2 !

[notdef]

and

= (vS + s1, s2, [notdef] [notdef] [notdef] , sN)T= (s1, s2, [notdef] [notdef] [notdef] , sN)T . (4.1)

Then the e ective potential takes a form

Ve = const. + 12m2hh2 +

2 vH h [notdef] +

H = 1

p2[parenleftBigg][parenleftBigg]

0 vH + h

[parenrightBigg]

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,

1

2m2s [notdef] +

3! vHh3 +

hhh

hhhh

4! h4

+ hss

4 h2 [notdef] +

hhss

ssss

4! ( [notdef] )2 + . . . , (4.2)

where only up to dimension-four interactions are shown explicitly.

We show the values of the coupling constants among the scalar particles in table 4. They correspond to the parameters of tables 2 and 3. In the case N = 1, the triple self-coupling of the Higgs boson hhh turns out to be larger than the SM value by a factor1.71.8, while the quartic self-coupling hhhh is larger by a factor 3.74.5. The range of each value shows the level of accuracy of our prediction. (In general higher derivatives of the potential have larger uncertainties.) These values are barely dependent on the input value for S, since the e ective potential in case (I) is independent of S on the -axis, and the dependence is weak in the cases (II) and (III).

For N = 1, the coupling constants involving the singlet scalars are large and of order ten. They become even larger if the input value for S is taken to be larger. One may wonder if perturbation theory is valid with such large coupling constants. We remind the reader that the coupling constants in the original Lagrangian are not so large, and that our predictions have been tested to be within the perturbative regime in the vicinity of the

16

scale of the vacuum. These large couplings are considered to be a typical feature of the present model and need to be tested by future experiments.

For larger N, the Higgs triple self-coupling is barely dependent on N, while the quartic coupling decreases slightly with N. The N dependences of the couplings involving the singlet scalars are more evident. Generally we obtain smaller couplings for larger N. This originates from high sensitivities of these couplings on HS, which reduces with N as

1/pN, and can be regarded as a characteristic feature of our potential.

In the cases (I) and (III) the values of ssss are not shown in the table, for the following

reason. If we compute the fourth derivative of VNLO with respect to and set h = [notdef][notdef] = 0,

the contribution of the NG modes of H diverges. This does not happen in the case (II) since VNLO is not involved. The divergence originates from an infra-red region, k 0 in

[integraltext]

d4k/(k2)2, and is an artifact of setting all the external momenta to zero. In physical amplitudes, momenta owing into the quartic vertex are almost always non-zero, which regularize the IR divergence in the loop integral. Since the other couplings have similar values for the cases (I), (II), (III), we expect that the values of ssss for the case (II) would

give reasonable estimates of the quartic vertex appropriate for physical amplitudes.

4.2.2 Comparison with results using Gildener-Weinbergs framework

Let us compare our results with those using the conventional GW framework.

We show in table 5(a)(c) comparisons of the couplings among the physical Higgs boson and singlet scalar(s), including their self-couplings, as dened in eq. (4.2). The e ective potential in the GW framework coincides with our e ective potential after setting ' = 0. Hence, the Higgs self-couplings are the same in both analyses. [We list the values for case(I) of our analysis as the corresponding values in the GW framework.]

In conventional analyses using the GW framework, the couplings involving the singlet

scalars are derived using the tree-level interactions (by substituting the VEV of the Higgs boson appropriately). Therefore, loop-induced e ects are not included. In our case, since the portal coupling HS is large, its loop-induced e ects can be large. These e ects are most enhanced for N = 1, where HS is the largest; see table 5(a). In particular there is a large enhancement of the singlet quartic self-coupling ssss, if S is small and the e ect of

HS-induced loop contribution is much larger. (We list the case S = 0.1 in the table.) Note that, although S is practically a free parameter, a smaller value is preferred with regard to the Landau pole. For completeness, we show the S dependences of our predictions in table 6. We see that only ssss depends considerably on S, and the dependence is roughly consistent with that of 6 S + const., as anticipated.

In view of the large portal coupling, certainly it is sensible to include the loop-induced e ects in computing the interactions involving the singlet scalars. In this case, we need to set up a formulation with proper account of order counting, as described in the previous section. It inevitably requires departure from the analysis in a one-dimensional subspace,i.e., the GW formulation.

The di erences between our results and those of the GW framework tend to decrease

for larger N, since the portal coupling becomes smaller. This tendency can be conrmed in tables 5(b)(c).

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N = 1 case hhh hhhh hss hhss ssss

SM prediction 0.78 0.78 none none none GWs framework 1.3 2.9 2 HS = 9.6 2 HS = 9.6 6 S = 0.6

(I) 1.3 2.9 11.4 13.8 our analysis (II) 1.4 3.4 10.2 13.0 6.5(III) 1.4 3.6 10.2 13.5

(a)

N = 4 case hhh hhhh hss hhss ssss

SM prediction 0.78 0.78 none none none GWs framework 1.3 2.9 2 HS = 4.8 2 HS = 4.8 6 S = 0.6

(I) 1.3 2.9 5.0 5.6 our analysis (II) 1.3 2.5 5.0 5.7 1.9(III) 1.4 2.7 5.0 5.7

(b)

N = 12 case hhh hhhh hss hhss ssss

SM prediction 0.78 0.78 none none none GWs framework 1.3 2.9 2 HS = 2.8 2 HS = 2.8 6 S = 0.6

(I) 1.3 2.9 2.8 3.0 our analysis (II) 1.3 2.2 3.0 3.2 0.92(III) 1.3 2.4 2.8 3.1

(c)

Table 5. Comparisons between the results derived by our analysis and those derived by GWs method for N = 1, 4 and 12 cases in (a), (b) and (c), respectively.

S 0.1 0.3 0.5 1.0 2.0 hss (I) 11.4 11.5 11.5 11.7 11.9

hss (II) 10.2 10.2 10.3 10.3 10.4 hss (III) 10.2 10.3 10.3 10.4 10.4 ssss (II) 6.53 7.91 9.32 12.8 19.7

Table 6. S dependences of the couplings involving the singlet scalar bosons.

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4.3 Veltmans condition for the Higgs mass

Veltmans condition is a condition for the quadratic divergence to vanish in the radiative correction to the Higgs potential. Generally the one-loop e ective potential for the Higgs boson in a cut-o regularization scheme is given by

V1-loop() = 1642 STr

4

ln 2 12[parenrightbigg]+ 2M2() 2 + M4()

ln M2()

2 12[parenrightbigg][bracketrightbigg]+ c.t. ,

(4.3)

where is a UV regulator, and M2()( 2) is the mass-squared matrix . The rst term

is a cosmological constant term, which we subtract by the counter term (with tremendous ne-tuning, but we will not be concerned about it here). The second term is a quadratically divergent term. It can either be subtracted (ne-tuning), which makes the theory unnatural, or vanish if

1

2

vector around the VEV [angbracketleft]

[vector]

[angbracketright] .

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[angbracketleft]

[vector]

[angbracketright] +

[vector]

[parenrightBig][bracketrightBig]

[vector]

!

[vector]0 = 0 for each i, j (4.4)

holds at a scale [notdef]0, which makes the theory natural. Here, [vector]

denotes a uctuation eld

@2 @( i)@( j)

hSTr M2

In the SM, the coe cient of the quadratically divergent term is given by

1

2

[notdef]=vH, h!0 =9g2(vH)4 +3g[prime]2(vH)4 6y2t(vH) + 6 H(vH)

[similarequal] 4.1 . (4.5)

Thus, Veltmans condition is violated at order unity and ne-tuning is necessary if the cut-o scale is much higher than the electroweak scale.

Here, we examine whether the ne-tuning is tamed comparatively in the scale-invariant model. The coe cient of 2 is expressed by the matrix

1

2

@2 @h2

STr M2 (vH + h)

@2 @( i)@( j)

hSTr M2

[angbracketleft]

[vector]

[angbracketright] +

[vector]

[parenrightBig][bracketrightBig]

[vector] ! [vector]0

=

0

@

9g2(vH)

4 + 3g

[prime]2(vH)

4 6y2t(vH) + 6 H(vH) + N HS(vH) 0

0 (4 HS(vH) + (N + 2) S(vH))1N[notdef]N

1

A.

(4.6)

It is diagonal, since there is no mixing between the Higgs boson and the singlet scalar boson. Let us moderately examine only the coe cient of the Higgs boson for various N. The hh component of the above matrix takes values 0.76, 5.6, 12.8 for N = 1, 4, 12, respectively. Instead, we can examine the constraint by Veltmans condition on the parameters of the model, see gures 6(a)(c), which also combine the phenomenologically valid regions plotted in gures 5. As can be seen, the ne-tuning is tamed particularly for N = 1, which may be a good feature of the model.11

11Since the cut-o scale, given by the Landau pole, is at several TeV scale in the N = 1 case, there will not be a serious ne-tuning problem below this scale.

19

(a) (b)

(c)

Figure 6. Favoured regions of Veltmans condition for the Higgs mass for N = 1 [(a)], N = 4[(b)] and N = 12 [(c)]. We combine the plot in each case with the respective plot of gures 5. The regions in the darkest, darker and light colors correspond to the magnitudes of the coe cient of the quadratic divergence [given by the hh component of eq. (4.6)] smaller than 0.1, 0.5 and 1.0, respectively.

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On the other hand, the quadratic divergence for the singlet component is signicant. Although S is a free parameter, it is expected to be positive semi-denite at the electroweak scale in order to stabilize the vacuum at LO.12 One way to remedy this problem is to couple right-handed Majorana neutrinos to the singlet scalars, as advocated in [11]. Here, we do not pursue such possibilities and leave it as an open question.

5 Conclusions and discussion

Up to now experimental data show that properties of the Higgs boson are consistent with the SM predictions. It is an intriguing question, with respect to the structure of the vacuum, whether only the Higgs self-interactions not measured so far can deviate signicantly from the SM predictions. A possible scenario is that the Higgs e ective potential is irregular at the origin. In this analysis we studied an extension of the Higgs sector with classical scale invariance as an example of such potentials. As a minimal model, we considered the SM Higgs sector in the scale-invariant limit, coupled via a portal interaction to an N-plet of real scalars under a global O(N) symmetry (singlet under the SM gauge group). We analyzed the electroweak symmetry breaking by the CW mechanism using RG and examined interactions among the scalar particles which probe the structure of the potential around the vacuum.

We computed the e ective potential in the conguration space of the Higgs and singlet scalar elds (, '). The input parameters are the Higgs self-coupling H, portal coupling HS, and self-coupling of the singlet scalars S at tree level. In order to obtain perturbatively valid predictions, we nd that parametrically [notdef] H[notdef] [notdef] HS[notdef] needs to be satised.

Furthermore, HS needs to be large, of order

pNC/N yt 1.7N1/2, in order to overwhelm the contribution of the top quark loop. We developed a special perturbative formulation for our model in order to analyze the (, ') space, since a naive treatment fails. The consistent parameter region of the couplings ( H, HS, S) is found which gives a global minimum of the e ective potential at (, ') = (vH, 0) with the Higgs VEV vH = 246 GeV and the Higgs boson mass mh = 126 GeV. The potential is stable against RG improvements, showing validity of the perturbative predictions. H and HS are almost xed, while S(> 0) can be taken fairly freely. The SM gauge group is broken as in the SM, while the O(N) symmetry is unbroken. (Hence, there is no NG boson.) The Higgs boson and singlet scalars do not mix at the vacuum. The mass of the N degenerate singlet scalars arises at LO of the perturbative expansion, whereas the mass of the Higgs boson is generated at NLO. Hence, generally the singlet scalars are much heavier than the Higgs boson. For N = 1 and S = 0.1 the mass is about 500 GeV, and it becomes lighter for larger N and heavier for larger S.

Since the coupling HS should be large in order to beat the top loop contribution, the Landau pole appears at a few to a few tens TeV (the position of the Landau pole is higher for larger N and smaller for larger S). Hence, the cut-o scale of this model is considered to be around this scale. For instance, this feature conicts with a scenario which imposes

12Since one-loop e ects induced by the portal coupling are large, there may be a consistent region where S is negative and still the vacuum is stabilized.

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a classically scale-invariant boundary condition at the Planck scale as a possible solution to the naturalness problem. Even without such a motivation, it can be a serious drawback of this model that the Landau pole is located so close to the electroweak scale. These features are consistent with the results of the analysis given in [9], while we presented more detailed analyses.

We computed the triple and quartic (self-)couplings of the Higgs and singlet scalar particles at the vacuum. Computation of the interactions involving the singlet scalars is a unique aspect of the use of our formulation, since the portal coupling is large and loop-induced e ects tend to be large. We obtain the Higgs triple and quartic self-couplings which are larger than the SM values by factors 1.61.8 and 2.84.5, respectively. The triple coupling is hardly dependent on N, while the quartic coupling decreases slightly with N. Both of them are barely dependent on S. According to the studies [35, 36], we naively expect that the triple coupling can be detected at 2 level at a future ILC, with an integrated luminosity of 1.1 ab1 at a centre-of-mass energy ps = 500 GeV and 140 fb1 at ps = 1 TeV. The couplings involving the singlet scalars are fairly large, of order ten, in the case of small N, reecting the irregularity of the potential at the origin and a large value of HS. If probed, these large couplings provide fairly vivid clues to the structure of the e ective potential in the vicinity of the vacuum.

It is di cult to detect signals of this model at the current LHC experiments. Since there is no mixing between the Higgs and singlet scalar particles, the couplings of the Higgs boson with other particles are unchanged from the SM values at tree level. The singlet scalar particles interact with the SM particles only through the Higgs boson. Furthermore, the singlet scalars are heavy and can be produced only in pairs due to the O(N) symmetry (or Z2 symmetry for N = 1). These features make it di cult to detect the singlet scalars directly at the LHC experiments. The production cross sections of the singlet scalars at the LHC are expected to be very small and it would be di cult to detect them, although a further detailed analysis is needed. On the other hand, it would be di cult to probe the extended Higgs sector from loop e ects in various precision measurements. These appear as higher-order e ects to the Higgs e ects, and since the Higgs e ects themselves are small, we expect that detection of an anomaly is non-trivial.

From the cosmological point of view, there is a possibility that the singlet scalar boson(s) is a part of dark matter. The singlet boson is stable due to O(N) or Z2 symmetry.

Since the couplings between singlet(s) and Higgs hss and hhss are very large, the anni

hilation cross section is also large, which decreases its relic abundance. In N > 1 cases, the total relic abundance is the sum over each individual singlet si. Then typically the scalar couplings decrease and the total relic abundance increases with N. Our naive estimation shows that singlet(s) can be a dark matter, whose abundance is less than around 1% of the total dark matter relic abundance [37]. We are currently preparing a further detailed study.

We also examined Veltmans condition (vanishing of the quadratic divergence) for the Higgs mass. We nd that for N = 1 the ne-tuning is relaxed compared to the SM, by the e ect of the large portal coupling which cancels against the top-quark loop e ects. Since the current level of the ne-tuning in the SM indicates that the natural scale of the cut-o

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is quite close to the electroweak scale, this may be a good tendency of the model. The relaxation of ne-tuning is not signicant for N > 1.

A possible scenario to avoid the Landau pole near the electroweak scale is to promote the global O(N) symmetry to a gauge symmetry (the group can also be replaced by another non-abelian group). An appropriate gauge-symmetric extension pushes up the location of the Landau pole, driven by the asymptotically-free nature, and in an extreme case, up to the Planck scale in the context of a radiative electroweak symmetry breaking scenario [9].

Since the Higgs self-interactions by higher powers of the Higgs eld are not suppressed and scalar interactions are large, one may suspect that our model belongs to one of the strongly-interacting Higgs sector, which can be analyzed, e.g. using a non-linear sigma model. We note that it is not a strongly-interacting model, at least around the electroweak scale. We reemphasize that our predictions are well within the perturbative regime, and the usual loop expansion with only renormalizable interactions gives stable predictions with only a few input parameters. To realize a phenomenologically valid scenario, a slightly unusual order-counting is employed, as discussed in section 3.

Acknowledgments

The works of K.E. and Y.S. were supported in part by Tohoku University Institute for International Advanced Research and Education and by Grant-in-Aid for scientic research No. 23540281 from MEXT, Japan, respectively.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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