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arXiv:0806.3190v3 [hep-ph] 22 Oct 2009

CERN-PH-TH/2008-058

On the origin of approximate custodial symmetry in

the Two-Higgs Doublet Model.

C. D. Froggatta,b, R. Nevzorova 1, H. B. Nielsenb, D. Thompsona

aDepartment of Physics and Astronomy,

Glasgow University, Glasgow, Scotland

bThe Niels Bohr Institute, Copenhagen, Denmark

Abstract

We argue that the consistent implementation of the multiple point principle (MPP)

in the general non-supersymmetric two Higgs doublet model (2HDM) can lead to a

set of approximate global custodial symmetries that ensure CP conservation in the

Higgs sector and the absence of flavour changing neutral currents (FCNC) in the

considered model. In particular the existence of a large set of degenerate vacua at

some high energy scale Λ caused by the MPP can result in approximate U(1) and

Z2symmetries that suppress FCNC and CP–violating interactions in the 2HDM.

We explore the renormalisation group (RG) flow of the Yukawa and Higgs couplings

within the MPP inspired 2HDM with approximate custodial symmetries and show

that the solutions of the RG equations are focused near quasi–fixed points at low

energies if the MPP scale scale Λ is relatively high. We study the Higgs spectrum

and couplings near the quasi–fixed point at moderate values of tanβ and compute

a theoretical upper bound on the lightest Higgs boson mass. If Λ ? 1010GeV

the lightest CP–even Higgs boson is always lighter than 125GeV. When the MPP

scale is low, the mass of the lightest Higgs particle can reach 180 − 220GeV while

its coupling to the top quark can be significantly larger than in the SM, resulting

in the enhanced production of Higgs bosons at the LHC. Other possible scenarios

that appear as a result of the implementation of the MPP in the 2HDM are also

discussed.

1On leave of absence from the Theory Department, ITEP, Moscow, Russia

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1 Introduction

The understanding of the origin of the strong suppression of flavour changing neutral

current (FCNC) transitions observed in Nature together with the origin of CP violation,

are among the major outstanding problems in particle physics. In the standard model

(SM) CP violation arises from the phase of the CKM matrix [1]-[2] and from the “θ-

term” in the QCD Lagrangian. Within the SM the particle content, gauge invariance

and renormalizability imply the absence of FCNC transitions at the tree level. At one–

loop, they are further suppressed by light quark masses (when compared to MW), i.e.

through the GIM mechanism [3], and by small mixing between the third and the first two

generations.

However because of the possible presence of new physics the SM should be regarded

as an effective “low energy” theory which, up to some scale Λ, is a good approximation

to the more fundamental underlying theory. Therefore the renormalizable interactions of

the SM are in general supplemented by higher dimensional interaction terms suppressed

by some powers of the scale Λ. These new interactions introduce new sources of CP

violation. In the considered case SU(3)C× SU(2)W× U(1)Y invariance is not sufficient

any more to protect the observed strong suppression of the FCNC processes. Under these

circumstances we may expect that either the scale Λ is huge1or dangerous new interactions

are absent because of symmetries of the underlying theory. If the only suppression of

FCNC processes is due to the scale Λ, then there is a tension between the new physics

scale which is required in order to solve the hierarchy problem and the one which is needed

in order to satisfy the experimental bounds from flavour physics. This is the so–called

new physics flavour problem [4].

In this article we consider the multiple point principle (MPP) [5]-[7] as a possible

mechanism for the suppression of the flavour changing neutral current and CP–violation

effects within the general non-supersymmetric two Higgs doublet extension of the SM

[8]–[9]. The violation of CP invariance and the existence of tree–level flavour–changing

neutral currents are generic features of SU(2)W×U(1)Y theories with two and more Higgs

doublets. Potentially large FCNC interactions appear in these models, because the diago-

nalization of the quark mass matrix does not automatically lead to the diagonalization of

the two or even more Yukawa coupling matrices, which describe the interactions of Higgs

bosons with fermionic matter. Moreover the Higgs potential of the two–Higgs doublet

model (2HDM) contains a lot of new couplings. Some of them may be complex, resulting

1The strongest bounds are obtained from K0− K0mixing and CP violation in K meson decay

measurements that forbid any Λ below 104TeV. The measurements of CP violation in B meson decay

as well as in D0− D0and B0− B0mixings imply that Λ ? 103TeV [4].

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in CP violation in the Higgs sector [10]-[15]. Although one can eliminate the violation

of CP invariance in the Higgs sector and tree–level FCNC transitions by imposing a dis-

crete Z2symmetry, such a symmetry leads to the formation of domain walls in the early

Universe [16] which create unacceptably large anisotropies in the cosmic microwave back-

ground radiation [17]. Therefore in practice it is necessary to impose only an approximate

symmetry, typically broken by soft mass terms.

The MPP postulates the existence of many phases with the same energy density

which are allowed by a given theory [5]-[7]. When applied to the SM, the multiple point

principle implies that the Higgs effective potential possesses two degenerate minima taken

to be at the electroweak and Planck scales respectively. The degeneracy of vacua at the

electroweak and Planck scales can be achieved only if (see [18])

Mt= 173 ± 4GeV,MH= 135 ± 9GeV. (1)

This MPP prediction for the Higgs mass lies on the SM vacuum stability curve [9], [19]–

[31] corresponding to the cut-off Λ = MPl

2. The hierarchy between the electroweak and

Planck scales might also be explained by MPP within the pure SM, if there exists a third

degenerate vacuum [32]-[34].

If we require the vacuum we live in to be just metastable w.r.t. decay into the sec-

ond vacuum, rather than being exactly degenerate with it, and otherwise make similar

assumptions to those in [18], the energy density in the second vacuum falls below that of

the vacuum in which we live. Consequently the Higgs mass is then predicted to be a bit

smaller. With the value used in this article for the top quark mass [35], Mt= 171.4±2.1

GeV, the value predicted for the Higgs mass from borderline metastability of our vacuum,

which we call meta-MPP [36], becomes MH= 118.4±5 GeV. This is remarkably close to

the two-standard deviation hint of a Higgs signal seen in LEP [37] at 115 GeV.

In previous papers [38]-[40] the MPP assumption has been adapted to models based

on (N = 1) local supersymmetry – supergravity, in order to provide an explanation for

the small deviation of the cosmological constant from zero. Recently we also considered

the application of the MPP to the SUSY inspired two Higgs doublet model of type II [41].

We established MPP conditions in this model and discussed the restrictions on the mass

of the SM–like Higgs boson caused by the MPP. Here we are going to extend this analysis

to the general 2HDM.

In the next section we specify the model. In section 3 we present the derivation of

the MPP conditions that result in approximate custodial U(1) and Z2symmetries. The

renormalisation group (RG) flow of Yukawa couplings within these MPP inspired two

2The requirement of the validity of perturbation theory up to the Planck scale leads to an upper

bound on MHin the SM, which is about 180 − 190GeV [19]-[23].

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Higgs doublet models is considered in section 4. In particular, we establish the positions

of quasi–fixed points and argue that the quasi–fixed point scenarios with large tanβ lead

to unacceptably large values of the top quark mass. In section 5 we study the evolution of

Higgs self–couplings and analyse the spectrum of Higgs bosons and their couplings near

the quasi–fixed point at moderate tanβ. We examine the phenomenological viability of

other possible MPP solutions in section 6. Our results are summarised in section 7. In

Appendix A the β–functions of Higgs self–couplings in the general two Higgs doublet

extension of the SM are presented. The derivation of the other MPP conditions that do

not give rise to an approximate custodial U(1) symmetry is discussed in Appendix B.

2 Two Higgs doublet extension of the SM

The most general renormalizable SU(2)W×U(1)Y gauge invariant potential of the model

involving two Higgs doublets is given by

Veff(H1,H2) = m2

1(Φ)H†

1H1+ m2

2(Φ)H†

2H2−

?

m2

3(Φ)H†

1H2+ h.c.

?

1H2|2

+

λ1(Φ)

2

?λ5(Φ)

(H†

1H1)2+λ2(Φ)

2

(H†

2H2)2+ λ3(Φ)(H†

1H1)(H†

2H2) + λ4(Φ)|H†

+

2

(H†

1H2)2+ λ6(Φ)(H†

1H1)(H†

1H2) + λ7(Φ)(H†

2H2)(H†

1H2) + h.c.

?

(2)

where

Hn=

χ+

n

(H0

n+ iA0

n)/√2

n = 1,2.

It is easy to see that the number of couplings in the two Higgs doublet model potential

compared with the SM grows from two to ten. Furthermore, four of them m2

3, λ5, λ6

and λ7can be complex, inducing CP–violation in the Higgs sector. In what follows we

suppose that the mass parameters m2

iand Higgs self–couplings λiof the effective potential

(2) only depend on the overall sum of the squared norms of the Higgs doublets, i.e.

Φ2= Φ2

1+ Φ2

2,Φ2

n= H†

nHn=1

2

?

(H0

n)2+ (A0

n)2

?

+ |χ+

n|2.

The dependence of m2

iand λion Φ is described by the renormalization group equations,

where the renormalization scale is replaced by Φ.

At the physical minimum of the scalar potential (2) the Higgs fields develop vacuum

expectation values

< H0

1>= v1, < H0

2>= v2

(3)

breaking the SU(2)W× U(1)Y gauge symmetry to U(1)emassociated with electromag-

netism and generating the masses of all bosons and fermions. The overall Higgs norm

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< Φ >=

?

|v1|2+ |v2|2

2

=

v

√2= 174GeV is fixed by the Fermi scale. At the same time

the ratio of the Higgs vacuum expectation values remains arbitrary. Hence it is convenient

to introduce tanβ = |v2|/|v1|.

As has been already mentioned in the Introduction, the Yukawa interactions of the

Higgs fields H1and H2with quarks and leptons generate phenomenologically unwanted

FCNC transitions. In particular these interactions contribute to the amplitude of K0−K

oscillations and give rise to new channels of muon decay like µ → e−e+e−. The common

way to suppress flavour changing processes is to impose a certain protecting custodial Z2

0

symmetry that forbids potentially dangerous couplings of the Higgs fields to quarks and

leptons [42]. Such a custodial symmetry requires the vanishing of the Higgs couplings λ6

and λ7. It also requires the down-type quarks to couple to just one Higgs doublet, H1say,

while the up-type quarks couple either to the same Higgs doublet H1(Model I) or to the

second Higgs doublet H2(Model II) but not both3. In fact, as we shall use in subsection

3.3, it is possible to generalise the idea of such a Z2symmetry so that each fermion couples

to just one Higgs field (H1or H2) but in a generation dependent way. The custodial Z2

symmetry forbids the mixing term m2

3(Φ)(H†

1H2) in the Higgs effective potential (2). But

usually a soft violation of the Z2symmetry by dimension–two terms is allowed, since it

does not induce Higgs–mediated tree–level flavor changing neutral currents (FCNC).

The set of RG equations that determines the running of Yukawa and Higgs couplings

in the two Higgs doublet model with exact and softly broken Z2symmetry can be found

in [43]–[47]. The constraints on the Higgs masses in the 2HDM with an unbroken Z2

symmetry have been examined in a number of publications [46]–[54]. The analysis of [54]

was performed assuming vacuum stability and the applicability of perturbation theory

up to a high energy scale (of order the grand unification scale), revealing that then all

Higgs boson masses lie below 200GeV. A very stringent restriction on the masses of

the charged and pseudoscalar states was found. They do not exceed 150GeV. However

such a light charged Higgs boson is ruled out by the direct searches for the rare B–meson

decays (B → Xsγ) in the Model II of the 2HDM, which cannot therefore be valid with

an unbroken Z2symmetry up to the unification scale. The theoretical restrictions on the

mass of the SM–like Higgs boson within the 2HDM with a softly broken Z2symmetry

were studied in [55].

We emphasize that, in this article, we do not impose any custodial symmetry but

rather consider the general Higgs potential (2). Instead we require that at some high

energy scale (MZ<< Λ ? MPl), which we shall refer to as the MPP scale Λ, a large set

3Similarly the leptons are required to only couple to one Higgs doublet, usually chosen to be the same

as the down-type quarks. However there are variations of Models I and II, in which the leptons couple

to H2rather than to H1.

4