# Realistic Composite Higgs Models

**ABSTRACT** We study the role of fermionic resonances in realistic composite Higgs models. We consider the low energy effective description of a model in which the Higgs arises as the pseudo-Goldstone boson of an SO(5)/SO(4) global symmetry breaking pattern. Assuming that only fermionic resonances are present below the cut-off of our effective theory, we perform a detailed analysis of the electroweak constraints on such a model. This includes the exact one-loop calculation of the T parameter and the anomalous Zbb coupling for arbitrary new fermions and couplings. Other relevant observables, like b to s gamma and Delta B=2 processes have also been examined. We find that, while minimal models are difficult to make compatible with electroweak precision tests, models with several fermionic resonances, such as the ones that appear in the spectrum of viable composite Higgs models from warped extra dimensions, are fully realistic in a large region of parameter space. These fermionic resonances could be the first observable signature of the model at the LHC. Comment: 33 pages, 8 figures

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arXiv:0901.2117v2 [hep-ph] 26 Mar 2009

Realistic Composite Higgs Models

Charalampos Anastasiou, Elisabetta Furlan, and Jos´ e Santiago

Institute for Theoretical Physics, ETH, CH-8093, Zurich, Switzerland

(Dated: March 26, 2009)

Abstract

We study the role of fermionic resonances in realistic composite Higgs models. We consider the

low energy effective description of a model in which the Higgs arises as the pseudo-Goldstone boson

of an SO(5)/SO(4) global symmetry breaking pattern. Assuming that only fermionic resonances

are present below the cut-off of our effective theory, we perform a detailed analysis of the electroweak

constraints on such a model. This includes the exact one-loop calculation of the T parameter

and the anomalous ZbL¯bL coupling for arbitrary new fermions and couplings.Other relevant

observables, like b → sγ and ∆B = 2 processes have also been examined. We find that, while

minimal models are difficult to make compatible with electroweak precision tests, models with

several fermionic resonances, such as the ones that appear in the spectrum of viable composite

Higgs models from warped extra dimensions, are fully realistic in a large region of parameter

space. These fermionic resonances could be the first observable signature of the model at the LHC.

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I.INTRODUCTION

One main objective of the Large Hadron Collider (LHC) is to discover the precise mecha-

nism of electroweak symmetry breaking (EWSB). A well motivated hypothesis is that there

exists a Higgs boson which is not a fundamental scalar. Instead it could be a composite

state of a strongly coupled theory, the pseudo-Goldstone boson of a spontaneously broken

global symmetry [1]. Compositeness can explain the insensitivity of EWSB to ultravio-

let physics, while the pseudo-Goldstone nature of the Higgs boson may explain the little

hierarchy between the scale of new physics and the scale of EWSB.

This mechanism has recently received increased attention, due to the realization that

calculable composite Higgs models can be constructed in five dimensions [2]. The main idea

is an old one [3], but only recently realistic models in warped extra dimensions [2, 4, 5, 6, 7]

have been constructed. The experience with five-dimensional models indicates that custodial

symmetry [8] and a custodial protection of the ZbL¯bLcoupling [9] are likely ingredients of

realistic constructions. Little Higgs models [10] also use the idea of a Higgs boson which is

the pseudo-Goldstone boson of a spontaneously broken global symmetry. In order to solve

the little hierarchy problem, they employ the mechanism of collective symmetry breaking,

which ensures that the Higgs mass remains insensitive to ultraviolet physics at one loop.

The main phenomenological implications of a Higgs boson which is the pseudo-Goldstone

boson of an extended broken symmetry are largely independent of the particular details of

how the global symmetry is broken. They can therefore be conveniently described using an

effective Lagrangian approach [11, 12]. A reasonable starting point is a symmetry breaking

pattern that includes custodial symmetry. A minimal example of such a pattern is given

by a global SO(5) symmetry broken at a scale f to its custodially symmetric subgroup

SO(4) [4, 5].

Using this effective Lagrangian approach, it was argued in [12] that composite Higgs

models with an SO(5)/SO(4) symmetry breaking pattern are difficult to make compatible

with electroweak and flavor precision data without introducing a substantial fine-tuning.

The argument was based on very minimal models, in which fermionic resonances did not

span full representations of the SO(5) group. In addition, the estimation of electroweak

observables was made neglecting contributions which are formally subleading, but can be

relevant in specific situations. Two recent works extended the analysis of [12] to models in

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which the fermionic composites span full representations of SO(5) [13, 14]. Although these

analyses differ in several aspects, like the degree of explicit SO(5) symmetry breaking and

the symmetry breaking patterns, their outcome is somewhat similar. Only a small region

of parameter space is allowed by electroweak precision data in models with no significant

fine-tuning and one set of fermionic composites spanning a vector representation of SO(5).

In this paper, we investigate thoroughly the viability of models with an SO(5)/SO(4)

symmetry breaking pattern. Our analysis extends previous works in two ways, by making

a careful computation of the effect of the new fermionic states on electroweak observables

and by considering the effect of an extended fermionic sector.

In our study of electroweak precision constraints we use an exact one-loop calculation of

the relevant electroweak observables. We do this in complete generality, and our analytic

formulae can be used in other models. In particular, our result for the anomalous ZbL¯bL

coupling is, to the best of our knowledge, the first complete calculation for an arbitrary

number of new quarks with generic couplings. We find that such an exact computation can

be important when formally subleading effects in commonly employed approximations are

enhanced.

We also examine the possibility of multiple sets of fermionic composites, departing from

minimal constructions.Our motivation is the model presented in [7], in which a five-

dimensional realization of a composite Higgs model with SO(5)/SO(4) symmetry breaking

pattern was shown to be fully compatible with electroweak precision tests, flavor observ-

ables, electroweak symmetry breaking and the observed dark matter relic abundance. The

non-minimal fermion sector of the model in [7] was indispensable in order to render its pre-

dictions compatible with experimental data. The goal of our article is to present an effective

four-dimensional description of composite Higgs models with a non-minimal fermionic con-

tent and discuss their electroweak and flavor constraints. We show that there are large

regions of parameter space in which composite Higgs models can provide a fully realistic

description of EWSB without fine tuning.

The paper is organized as follows. In section II we briefly review the effective description

of composite Higgs models with an SO(5)/SO(4) symmetry breaking pattern, including the

experimental constraints of minimal models. Section III is devoted to a description of the

relevant fermionic sector of the theory and its effects on electroweak precision observables

and flavor physics. In section IV we present exact one-loop expressions for electroweak

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and flavor precision observables which are valid in general extensions of the SM. The main

phenomenological implications of our model are discussed in section V, where we describe two

options for realistic composite Higgs models, a very simple one and a slightly more involved

one which is closer to the realistic examples we know from extra dimensions. Finally, we

conclude in section VI.

II.EFFECTIVE DESCRIPTION OF COMPOSITE HIGGS MODELS

The low energy effective description of a composite Higgs model with SO(5)/SO(4) sym-

metry breaking pattern can be described by a scalar φ in the fundamental representation of

SO(5), subject to the constraint

φ2= f2,(1)

where f is the scale of the global symmetry breaking, assumed to be somewhat larger than

the EWSB scale v ≈ 174 GeV. The first four components of φ, which transform as the

fundamental representation of SO(4), are denoted by?φ. The SU(2)L× U(1)Y subgroup

of SO(4) = SU(2)L× SU(2)R, where the hypercharge corresponds to the T3

weakly gauged.1The vacuum expectation value of?φ breaks the EW symmetry,

Rgenerator, is

m2

W=g2v2

2

,v2=1

2??φ2?.(2)

From Eq. (1) we see that the ratio

sα≡ sinα ≡

√2v

f,

(3)

measures the Higgs compositeness, i.e. how the vev of φ is split between?φ and φ5. Canonical

normalization of the different components in φ, expanded around its vev, requires a rescaling

of the physical Higgs

h → cosα h ≡ cαh,(4)

whereas the would be Goldstone bosons are not modified. This redefinition implies an

important feature of Higgs compositeness, namely that Higgs couplings to gauge bosons

are suppressed with respect to the couplings in the SM by the factor cα=

?1 − 2v2/f2.

1An extra U(1) group is required to generate the correct Weinberg angle, but it is irrelevant for the present

discussion and will be disregarded.

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In the case of fermions, the suppression factor depends also on the embedding of the SM

fermions in SO(5) representations [6, 15]. This suppression of the Higgs couplings affects the

quantum corrections to electroweak precision observables, leading to some sensitivity to the

ultraviolet cut-off [12]. The leading effect can be taken into account by replacing the Higgs

mass with an effective (heavier) Higgs in the SM expressions of the one-loop corrections to

the electroweak precision observables,

mEWPT,eff= mh(Λ/mh)s2

α, (5)

where Λ = 4πf/√NG= 2πf is the ultraviolet cut-off of our effective theory, NGbeing the

number of Goldstone bosons. This modification gives rise to an additional contribution to

the Peskin-Takeuchi [16] S and T parameters

∆S =

1

12πln

?

m2

EWPT,eff

m2

h,ref

?

,∆T = −

3

16πc2

W

ln

?

m2

EWPT,eff

m2

h,ref

?

,(6)

where mh,refis the reference Higgs mass used in the electroweak fit and cW is the cosine of

the Weinberg angle.

Furthermore, custodial symmetry can naturally account for a suppressed contribution of

ultraviolet physics to the T parameter, but there is no reason not to expect a contribution

to the S parameter from higher dimensional operators. A reasonable estimate, assuming

that new physics couples linearly to the SM (otherwise this estimate should have an extra

loop suppression), is

∆SΛ∼4s2

W

αem

g2v2

Λ2≈ 0.16

?3 TeV

Λ

?2

.(7)

The combination of the two corrections, (6) and (7), results in a positive shift to the

S parameter and a negative shift to the T parameter. We show in Fig. 1 the current

constraints on the S and T parameters, assuming a Higgs mass mh= 120 GeV. We also

show the contributions from Higgs compositeness and UV physics for different values of sα.

These constraints are obtained as follows. For reasons that will become apparent in the next

section, we have performed a fit to all the relevant electroweak observables, allowing the S

and T parameters and an anomalous coupling of the bLquark to the Z, that we denote by

δgbL, to vary (note that we have set U = 0 in our fit, as it is expected to be vanishingly

small in our model).2In Fig. 1, we have fixed the optimal value of δgbL= −2.5 × 10−4,

2We use Ref. [17] for the fit updated to the most recent experimental data [18]. αs(mZ) = 0.1183 and

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