# Buried Higgs boson

**ABSTRACT** We present an extension of the MSSM where the dominant decay channel of the Higgs boson is a cascade decay into a four-gluon final state. In this model the Higgs is a pseudo-Goldstone boson of a broken global symmetry SU(3)-> SU(2). Both the global symmetry breaking and electroweak symmetry breaking are radiatively induced. The global symmetry breaking pattern also implies the existence of a light (few GeV) pseudo-Goldstone boson eta which is a singlet under the standard model gauge group. The h -> eta eta branching fraction is large, and typically dominates over the standard h -> b b decay. The dominant decay of eta is into two gluons, while the decays to photons, taus or lighter standard model flavors are suppressed at the level of 10^(-4) or more. With h-> 4 jets as the dominant decay, the Higgs could be as light as 78 GeV without being detected at LEP, while detection at the LHC is extremely challenging. However many of the super- and global symmetry partners of the standard model particles should be easily observable at the LHC. Furthermore, the LHC should be able to observe a "wrong Higgs" that is a 300-400 GeV heavy Higgs-like particle with suppressed couplings to W and Z that by itself does not account for electroweak precision observables and the unitarity of WW scattering. At the same time, the true Higgs is deeply buried in the QCD background.

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**ABSTRACT:**This review provides an elementary discussion of electroweak symmetry breaking in the minimal and the next-to-minimal supersymmetric models with the focus on the fine-tuning problem — the tension between natural electroweak symmetry breaking and the direct search limit on the Higgs boson mass. Two generic solutions of the fine-tuning problem are discussed in detail: models with unusual Higgs decays; and models with unusual pattern of soft supersymmetry breaking parameters.Modern Physics Letters A 11/2011; 24(21). · 1.34 Impact Factor - SourceAvailable from: Zhen LiuDavid Curtin, Rouven Essig, Stefania Gori, Prerit Jaiswal, Andrey Katz, Tao Liu, Zhen Liu, David McKeen, Jessie Shelton, Matthew Strassler, Ze'ev Surujon, Brock Tweedie, Yi-Ming Zhong[Show abstract] [Hide abstract]

**ABSTRACT:**We perform an extensive survey of non-standard Higgs decays that are consistent with the 125 GeV Higgs-like resonance. Our aim is to motivate a large set of new experimental analyses on the existing and forthcoming data from the Large Hadron Collider (LHC). The explicit search for exotic Higgs decays presents a largely untapped discovery opportunity for the LHC collaborations, as such decays may be easily missed by other searches. We emphasize that the Higgs is uniquely sensitive to the potential existence of new weakly coupled particles and provide a unified discussion of a large class of both simplified and complete models that give rise to characteristic patterns of exotic Higgs decays. We assess the status of exotic Higgs decays after LHC Run 1. In many cases we are able to set new nontrivial constraints by reinterpreting existing experimental analyses. We point out that improvements are possible with dedicated analyses and perform some preliminary collider studies. We prioritize the analyses according to their theoretical motivation and their experimental feasibility. This document is accompanied by a website that will be continuously updated with further information: http://exotichiggs.physics.sunysb.edu.Physical Review D 12/2013; · 4.86 Impact Factor - SourceAvailable from: Javi Serra
##### Article: Composite Higgses

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**ABSTRACT:**We present an overview of composite Higgs models in light of the discovery of the Higgs boson. The small value of the physical Higgs mass suggests that the Higgs quartic is likely loop generated, thus models with tree-level quartics will generically be more tuned. We classify the various models (including bona fide composite Higgs, little Higgs, holographic composite Higgs, twin Higgs and dilatonic Higgs) based on their predictions for the Higgs potential, review the basic ingredients of each of them, and quantify the amount of tuning needed, which is not negligible in any model. We explain the main ideas for generating flavor structure and the main mechanisms for protecting against large flavor violating effects, and present a summary of the various coset models that can result in realistic pseudo-Goldstone Higgses. We review the current experimental status of such models by discussing the electroweak precision, flavor and direct search bounds, and comment on UV completions and on ways to incorporate dark matter.European Physical Journal C 05/2014; 74(2766). · 5.44 Impact Factor

Page 1

arXiv:0906.3026v2 [hep-ph] 22 Sep 2009

Buried Higgs

Brando Bellazzinia, Csaba Cs´ akia, Adam Falkowskib, and Andreas Weilerc

aInstitute for High Energy Phenomenology

Newman Laboratory of Elementary Particle Physics

Cornell University, Ithaca, NY 14853, USA

bNHETC and Department of Physics and Astronomy

Rutgers University, Piscataway, NJ 088550849, USA

cCERN Theory Division, CH-1211 Geneva 23, Switzerland

b.bellazzini@cornell.edu, csaki@cornell.edu, falkowski@physics.rutgers.edu,

andreas.weiler@cern.ch

Abstract

We present an extension of the MSSM where the dominant decay channel of the Higgs

boson is a cascade decay into a four-gluon final state. In this model the Higgs is

a pseudo-Goldstone boson of a broken global symmetry SU(3) → SU(2). Both the

global symmetry breaking and electroweak symmetry breaking are radiatively induced.

The global symmetry breaking pattern also implies the existence of a light (few GeV)

pseudo-Goldstone boson η which is a singlet under the standard model gauge group.

The h → ηη branching fraction is large, and typically dominates over the standard

h → bb decay.

photons, taus or lighter standard model flavors are suppressed at the level of 10−4

or more. With h → 4 jets as the dominant decay, the Higgs could be as light as

78 GeV without being detected at LEP, while detection at the LHC is extremely

challenging. However many of the super- and global symmetry partners of the standard

model particles should be easily observable at the LHC. Furthermore, the LHC should

be able to observe a “wrong” Higgs that is a 300-400 GeV heavy Higgs-like particle

with suppressed couplings to W and Z that by itself does not account for electroweak

precision observables and the unitarity of WW scattering. At the same time, the true

Higgs is deeply buried in the QCD background.

The dominant decay of η is into two gluons, while the decays to

Page 2

Introduction

The elegant and appealing idea of low-energy supersymmetry faces technical difficulties when

confronted with experimental data. In the MSSM at tree level the mass of the Higgs boson

is bounded from above by the Z boson mass. In order to reconcile the MSSM with the non-

discovery of the Higgs at LEP one has to assume that supersymmetry breaking introduces

large loop corrections to the Higgs quartic self-coupling (which sets the Higgs boson mass).

However, the same large parameters that lift the Higgs boson mass also contribute to the W

and Z mass (that is to the Higgs VEV), and large accidental cancellations are required to

keep the Z-mass at its experimental value. In a typical scenario this leads to a fine-tuning at

the level of 1% or worse. Additional theoretical structures beyond those of the MSSM are

necessary if supersymmetry is to provide a solution to the naturalness problem. The most

popular approach is to engineer new tree-level contributions to the quartic Higgs potential,

which is possible e.g in singlet (NMSSM) [1] or gauge [2] extensions of the MSSM.

Another interesting possibility for improving the naturalness of low-energy supersym-

metry is to assume that the Higgs is a pseudo-Goldstone boson (pGB) of a spontaneously

broken approximate global symmetry, much as in Little Higgs models [3], but now in a super-

symmetric context. This approach [4–10] is referred to as the double protection of the Higgs

potential or as the super-little Higgs1. The collective symmetry breaking pattern of little

Higgs models combined with supersymmetry indeed implies that one-loop corrections to the

Higgs mass are completely finite, thereby greatly reducing one-loop corrections to the W and

Z mass, and improving on naturalness. Unfortunately, due to the reduced sensitivity of the

Higgs potential to loop effects it is quite difficult to obtain a Higgs heavier than 114.4 GeV

in these models. Thus, additional (sometimes truly baroque) structures have to be invoked

to make the Higgs heavy enough, with the simplest complete model involving a U(1) gauge

extension [10].

However the Higgs does not necessarily have to be as heavy as 115 GeV if it decays in

a non-standard way. Dermisek and Gunion [13] (see also [14] and [15]) pointed out that

the LEP bounds can be relaxed if the Higgs undergoes a cascade decay to a many particle

final state, rather than directly decaying into a pair of SM particles. This is possible if for

example the Higgs can first decay to a pair of light singlet pseudoscalars, each of which

subsequently decays to a pair of quarks or leptons. Depending on the dominant decay

channels of the pseudoscalar the LEP bound on the Higgs mass can be as low as 86 GeV for

a 4τ final state [16], or almost as high as the standard bound 110 GeV for 4b final states [16]

(assuming the standard model (SM) production cross section for the Higgs), see [17] for

a review. The cascade decays of the Higgs are most often realized in the context of the

NMSSM [18], but there exist also non-supersymmetric realizations [19,20]. A Higgs boson

mass below 100 GeV would greatly reduce the number of problems that are plaguing model

building and could eradicate the little hierarchy problem.

The main point of this paper is to argue that the idea of Dermisek and Gunion is very

naturally realized in the double protection models: all the necessary ingredients are already

in there. Since the simplest models of that type are based on the SU(3) → SU(2) pattern

1For early attempts of supersymmetric theories with a pGB Higgs see [11,12]

1

Page 3

of spontaneous global symmetry breaking one ends up with five pGBs. Four of these are

identified with the Higgs doublet, while the remaining one (which is henceforth referred to

as η) is a singlet under the SM gauge interactions, and naturally has the properties of the

Dermisek-Gunion pseudoscalar needed to hide the Higgs at LEP! The singlet η receives a

mass from one-loop corrections which naturally fall into the few GeV range, thus allowing

the h → 2η decay. Furthermore, a pair of the pGB singlets has a higher-derivative tree-level

interaction vertex with the Higgs boson that is suppressed by the global symmetry breaking

scale f, which allows the Higgs cascade decays to have a large branching fraction. For

typical parameters of the model when the scale f is not much larger than the electroweak

scale, the branching ratio for the two-body Higgs decays to the SM fermions is less than

20% (see Fig.1). This is enough to hide an 80-100 GeV Higgs from LEP2provided the pGB

singlet is lighter than 9.2 GeV and therefore cannot decay to b-quarks. To calculate the

η mass one needs to specify the fermion structure of the theory. We show that with the

same matter content as [10], a large fraction of the η masses lies naturally in the interesting

range below 9.2 GeV. The actual Higgs phenomenology is then determined by the leading

decays of η. We argue that in our model the gluon decay channel is by far the dominant one,

resulting in the h → 4j signal that is in fact invisible at LHC due to the huge background.

Thus, double protection predicts very peculiar experimental signatures where a host of super-

and little partners are visible at the LHC but the Higgs boson responsible for electroweak

breaking remains elusive. However, a “wrong” Higgs may show up more easily at the LHC:

the theory predicts a heavy (300-400 GeV) scalar particle, corresponding to the oscillations

of the global symmetry breaking scale f, whose production cross section is 15-25% that of

the SM Higgs boson. This radial mode is similar to the Higgs in many respects, but it has

reduced couplings to W and Z gauge bosons, thus being unable on its own to account for

the electroweak precision fits and unitarization of gauge boson scattering amplitudes.

The paper is organized as follows: first we introduce the gauge and global symmetry

breaking structure and identify the Goldstones in the theory. We then calculate the h → 2η

branching fraction and compare it to the leading SM h → 2b channel. We specify the Yukawa

structure of the theory and we calculate the η mass. We present the distribution of the η

and Higgs masses and the necessary fine tuning needed to achieve those values. Then we

show that η → gg is the dominant decay channel, resulting in the Higgs decaying into 4 jets.

Finally we discuss the impact of this model on the electroweak precision observables and on

unitarity, and then conclude.

Gauge and Global Symmetries, Goldstones

The model we consider is based on the SU(3)C×SU(3)W×U(1)Xgauge symmetry [5–7,9,10]

which is the supersymmetric version of the simplest little Higgs model of [22]. SU(3)W×U(1)

is broken by two vectorlike sets Φu,dand Hu,dof Higgs superfields with the following quantum

2It may also provide the explanation of the large excess of Higgs like events seen at LEP for a Higgs mass

of about 98 GeV [21].

2

Page 4

numbers

SU(3)C

1

1

SU(3)W

3

¯3

U(1)X

1/3

−1/3

Hu,Φu

Hd,Φd

(1)

The central assumption here is that there are no cross-couplings between the two sets of

Higgses, that is the mass terms ΦdHuand HdΦuin the superpotential are absent or very

suppressed even though they are allowed by the gauge symmetry. It should be stressed

that this assumption is technically natural in the supersymmetric context thanks to the

non-renormalization theorems. The consequence of this assumption is that there is an ap-

proximate SU(3)1×SU(3)2global symmetry (broken to the diagonal SU(3)Wby the gauge

interactions) acting separately on the two Higgs sets. Φu,dare assumed to have a large VEV

(generated by some additional supersymmetry preserving dynamics) in the 10 TeV regime,

?Φu?T= ?Φd? = (0,0,F/√2).

This VEV determines the orientation of the SM group within the SU(3)×U(1)Xand breaks

the gauge group down to SU(3)C× SU(2)W× U(1)Y. In our convention SU(2)W is acting

on the upper two components of the SU(3)W triplets and hypercharge is realized as Y =

T8/√3+X with T8=

with all the Goldstone bosons being eaten by the SU(3)×U(1)/SU(2)×U(1) massive gauge

fields, but it leaves SU(3)2intact. The other set of Higgs triplets Hu,dis assumed to get

much smaller VEVs in the f = 300 − 500 GeV range (generated radiatively, much like the

Higgs VEV in the MSSM). This VEV will also break the SU(3)×U(1) gauge symmetry to

SU(2)×U(1) and produce its own 5 Goldstone bosons. If the two sets of VEVs are somewhat

misaligned then the only remaining unbroken gauge group is U(1)em, with the misalignment

responsible for electroweak breaking. In the limit F ≫ f the misalignment between the two

sets of VEVs is parameterized by the 5 Goldstone bosons as

(2)

1

2√3diag(1,1,−2). The VEV also breaks the SU(3)1global symmetry

Hu= eiΠ/ff sinβ

0

0

1

, Hd= e−iΠ/ff cosβ

0

0

1

, (3)

where the pion matrix corresponds to the broken generators

Π =

H

H†

˜ η

√2

. (4)

3 of the 5 Goldstones are eaten by the W and Z bosons after electroweak symmetry breaking,

with two real physical pGBs˜h, ˜ η remaining in the physical spectrum (the tilde here is to

stress that the field is not canonically normalized; the properly normalized fields h, η will

be defined below). In terms of the physical Goldstones the parametrization of the triplets is

3

Page 5

given by3

Hu= f sinβ

0

sin(˜h/√2f)

ei˜ η/√2fcos(˜h/√2f)

HT

d= f cosβ

0

sin(˜h/√2f)

e−i˜ η/√2fcos(˜h/√2f)

. (5)

The real field˜h is the pGB Higgs boson whose VEV breaks the electroweak symmetry. The

electroweak scale vEW= 174 GeV is related to the Higgs VEV ?˜h? =√2˜ v by

vEW= f sin(˜ v/f).(6)

The other pGB field η lives fully in the third component of the triplet, therefore it is a

perfect singlet under the SM gauge interactions. Thus there are no constraints on η from a

contribution to the Z-width.4

Higgs decays: h → 2η vs. h → b¯b

Next we discuss the Higgs decay modes, and argue that there is a possibility for the h → ηη

mode to dominate for generic values of the parameters. Even though η is an SU(2) singlet,

it does have a tree-level derivative coupling to the Higgs field h due to h partly living in

the third component of Hu,d(and not because of η mixing into the doublet part of Hu,d).

The symmetry preserving derivative coupling (characteristic to exact Goldstone bosons)

originates from the Higgs kinetic terms,

LpGB≈1

2(∂µ˜h)2+1

2cos2(˜h/√2f)(∂µ˜ η)2.(7)

At one loop there are also non-derivative interactions via the Coleman-Weinberg potential

that depend on both h and η, but these lead to subleading interaction terms. After the

Higgs gets a VEV we define the canonically normalized Higgs boson field h and the singlet

field η by˜h =

boson with two singlets:

Lhη2 ≈ −h(∂µη)2tan(˜ v/f)

√2˜ v + h, ˜ η = η/cos(˜ v/f). This leads to the following vertex of the Higgs

√2f

. (8)

The decay width of the Higgs boson into two singlets is given by

Γh→ηη≈

1

64π

?

1 −v2

EW

f2

?−1m3

hv2

f4

EW

.(9)

The coupling of the Higgs boson to the SM quarks and leptons is the same as in the SM, up

to an additional factor cos(˜ v/f) that arises due to its pGB nature. Thus, the decay width

3Taking into account 1/F corrections there is also a small component of the physical pGBs embedded in

Φu,d.

4Once η gets a mass there will be a small mixing with the physical pseudoscalar A living in the Higgs

doublets, but the mixing angle is suppressed by m2

η/m2

A

<

∼10−5.

4

Page 6

into a pair of SM fermions is given by

Γh→ff=

?

1 −v2

EW

f2

?

ΓSM

h→ff= cQCD

Nc

16π

?

1 −v2

EW

f2

?mhm2

v2

f

EW

.(10)

Here, Nc= 3 for quarks and 1 for leptons. cQCDarises due to higher order QCD corrections

which can be numerically important; for example for the b-quark it is given by cQCD≈ 1/2

for a 100 GeV Higgs [23]. The relevant quantity for LEP searches, customarily denoted as

ξ2BR(h → bb), is the branching ratio for a decay into b quarks multiplied by the suppression

of the Higgs production cross section. The latter should also be taken into account in our

model because the coupling of the pGB Higgs to the Z boson is suppressed, much as the

Higgs-fermion coupling, by a factor cos(˜ v/f). It then follows

ξ2BR(h → bb) ≡

σ(e+e−→ Zh)

σSM(e+e−→ Zh)BR(h → bb) =

ΓSM

h→bb

1 −Γh→ηη+

?

v2

EW

f2

??

fΓSM

h→ff

?

1 −v2

EW

f2

?2

(11)

We plot ξ2BR(h → bb) as a function of the Higgs mass for several choices of the global

symmetry breaking scale f, together with the combined LEP bound on ξ2BR(h → bb)

from [16]. If f is as small as 350 GeV, the bb branching ratio is sufficiently suppressed to

allow for a Higgs as light as the Z boson. Once f is raised to around 450 GeV or higher, the

generic 114.4 GeV limit from LEP cannot be significantly relaxed - the bb branching ratio

becomes large enough to have been observable at LEP.

Matter Yukawas

Of course, in order to make the Higgs decay into η one has to ensure that the latter is light

enough. Thus we now turn to discussing how the η mass is generated. At tree level the

η mass as well as the masses of the remaining four pGBs vanish, but they are generated

at one loop. The leading contributions are expected from the third generation quarks and

their symmetry partners (superpartners and global symmetry partners). Following [10], we

consider the following embedding of the third generation quarks and leptons into SU(3)C×

SU(3)W× U(1)Xrepresentations

SU(3)C

Q = (tQ,bQ,ˆbQ)

V = (bV,tV,ˆtV)

Vc= (bV

tc

b1,2

c

L1,2= (τL

Ec= (νE

ν1,2,3

c

SU(3)W

3

¯3

3

1

1

¯3

¯3

1

U(1)X

0

1/3

−1/3

−2/3

1/3

−1/3

2/3

0

3

3

¯3

¯3

¯3

1

1

1

c,tV

c,ˆtV

c)

1,2,νL

c,τE

1,2, ˆ νL

c, ˆ τE

1,2)

c)

(12)

5

Page 7

f ? 350GeV

f ? 400GeV

f ? 450GeV

8085 90 95

mh?GeV?

100 105 110

0.10

0.50

0.20

0.30

0.15

0.70

Ξ2Br?h?bb?

10

-2

10

-1

1

20 40 6080

mH(GeV/c2)

100120

95% CL limit on ξ2B(H→bb

–)

LEP

√s = 91-210 GeV

H→bb

–

(b)

Figure 1: The parameter ξ2BR(h → bb) in this model for 3 representative values of the

global symmetry breaking scale as a function of the Higgs mass. The dashed line is an

approximation of the observed LEP bound transcribed from the actual LEP plot reprinted

from [16] on the right hand side. We can see that while for f = 450 GeV the bound is over

110 GeV for the Higgs mass, for f = 350 GeV the Higgs could be lighter than 90 GeV.

This assignment of representations is anomaly free. The quark and lepton masses originate

from the Yukawa couplings and the supersymmetric mass terms

W = y1tcV Φu+y2HuVcQ+µVV Vc+yb1ΦdQb1

More Yukawa and mass terms are needed to give masses to all neutrinos but we are not

concerned with it here. As in the case of the triplet Higgs superpotential, these are not the

most general Yukawa couplings consistent with the gauge symmetries, in particular ˜ y1tcV Φu

and ˜ y2HuVcQ are omitted. Omitting those and other allowed terms amounts to imposing

a collective breaking of the global SU(3)2symmetry which acts on Hu,dand remains after

gauge symmetry breaking via Φu,dVEVs. Note that in the top sector SU(3)2is restored

if any of the three couplings: y1, y2or µV is set to zero. At the same time, in the bottom

sector SU(3)2is restored if either y2or yb1vanishes. The latter means that the bottom loops

induce corrections of order y2

M2

one value of yb1starts reintroducing the little hierarchy problem due to the large log(yb1F),

and in the following we assume yb1< 0.1 to keep fine-tuning under control. On the other

hand, in the top sector all Yukawa couplings can be order one, as long as µV is less than

TeV.

c+yb2HdQb2

c+yτ1ΦdL1Ec+yτ2HdL2Ec. (13)

2y2

b1log

yb1F

softto the Higgs mass. Since F ∼ 10 TeV, an order

Radiative symmetry breaking and fine tuning

The top Yukawa and mass terms included in eq. (13) at one-loop lead to radiative generation

of the global symmetry breaking scale f and the electroweak scale vEW. The former arises

6

Page 8

as a consequence of the negative contributions to the mass and quartic terms of the triplet

Hu,

≈ −3y2

2π2

3y4

8π2

m2

Hu

2sin2β

M2

softlog(Λ/MT),

λHu

≈

2sin4β

log((M2

soft+ M2

T)/M2

T),(14)

where MT=

the soft supersymmetry breaking scale (we assumed the common soft mass for all the stops

and F ≫ f). The potential (14) also generates the mass m2

of the triplet Hucorresponding to the fluctuations of the VEV f. This part of the potential

is in many respects similar to generating the Higgs potential in the MSSM. In particular, the

mass term is logarithmically divergent and proportional to the soft supersymmetry breaking

scale. Yet it does not lead to the fine-tuning problem at the same level as in the MSSM.

This is because 1) the scale f is larger than the electroweak scale 2) we are free to take

the Yukawa coupling y2to be larger than the SM top Yukawa coupling. One can define the

amount of fine tuning necessary to maintain the hierarchy between F and f as the ratio of

the physical radial mass to the loop induced correction of the triplet

?µ2

V+ sin2βy2

2f2is the mass of the heavy fermionic top partner, and Msoftis

r∼ 4λHuf2for the radial mode

FT3=m2

r/2

Hu|∼

|m2

y2

M2

2f2

soft

log

M2

soft+M2

M2

T

logΛ2

M2

T

T

.(15)

For example, for f ∼ 350 GeV and y2∼ 1.8 the fine-tuning is usually in the 5-10% range

and the couplings remain perturbative up to Λ ≈ 103− 104TeV. Note however, that the

entire low-energy theory below F could have been defined without actually specifying the

structure of the UV completion of the theory around F and the origin of the scale f. We

find it very appealing that such a simple theory perturbative up to large scales can be found.

It is entirely possible that other UV completions with even less tuning can give the same

low-energy physics around the TeV scale, for example a somewhat different anomaly free

fermion matter content can be also used [24].

The one-loop contributions to the pGB Higgs potential, on the other hand, are completely

finite and calculable. Electroweak symmetry breaking is triggered by negative contributions

to the Higgs mass parameter from top/stop loops,

∆m2≈ −

3m2

8π2v2

t

EW

?

M2

TlogM2

soft+ M2

M2

T

T

+ M2

softlogM2

soft+ M2

M2

soft

T

?

,(16)

while the contributions from the bottom sector are down by m2

one-loop contributions to the pGB Higgs quartic, and the Higgs boson mass is

b/m2

t≪ 1. There are also

m2

h

=

?

−2M2

1 −v2

EW

f2

??

m2

Zcos2(2β) +

3m4

4π2v2

t

EW

?

log

?

M2

softM2

soft+ M2

T

m2

t(M2

T)

?

soft

M2

T

log

?M2

soft+ M2

M2

soft

T

???

.(17)

7

Page 9

1.0 1.21.4 1.61.8 2.0

40

50

60

70

80

90

y2

mh

F ? 10TeV Msoft?700GeV y1? 0.5 yb1? 0.04

1.01.21.4 1.6 1.82.0

100

150

200

250

300

350

400

y2

mr

F ? 10TeV Msoft?700GeV y1? 0.5 yb1? 0.04

Figure 2: The mass of the pGB Higgs (left panel) and the radial mode (right panel) for a

sample slice of the parameter space, for f = 350 (dahshed blue) and f = 400 (solid red)

GeV. This plot was obtained using the full 1-loop Coleman-Weinberg potential including the

mixing between the Higgs and the radial mode.

Note that the tree-level Higgs boson mass is suppressed with respect to mZby the factor of

cos(˜ v/f) which is of order 0.8 − 0.9 for the interesting range of f. The one-loop corrections

lift the Higgs boson mass above the tree-level value, but for natural values of the heavy top

and soft mass they cannot add much more than 10 GeV. As a consequence, the Higgs mass

typically ends up in the 80-100 GeV range. While a complete 1-loop analysis of the full

spectrum is beyond the scope of this paper (for instance we have not included the heavy

Higgs scalars and the second radial mode rd), we have numerically calculated the 1-loop

effective Coleman-Weinberg potential for the light Higgs h and the radial mode r due to

the top-stop and bottom-sbottom loops. We have minimized this potential in the presence

of the tree-level SU(2)×U(1) D-terms explicitely breaking the global symmetry and also

soft breaking scalar masses. The resulting Higgs and radial masses for a typical choice of

parameters is displayed in Fig. 2. The fine tuning in the doublet Higgs potential (which is

usually the main source of fine tuning in the MSSM) defined as

FT2=m2

h/2

|∆m2|

(18)

is typically completely absent in that Higgs mass range. In summary, in absence of addi-

tional theoretical structures such as additional gauge singlet superfields or U(1) D-terms,

double protection combined with naturalness predict that the Higgs boson mass should be

kinematically accessible at LEP energies.

8

Page 10

η mass

Fortunately, double protection in its simplest version also predicts the existence of the singlet

pGB η via which the Higgs can cascade decay, thus avoiding discovery at LEP. The same

structure that ensures double protection of the pGB Higgs potential also ensures that the

singlet mass is much smaller than the Higgs boson mass. Indeed, in the limit of collective

symmetry breaking and the gauge symmetry breaking scale F going to infinity, the singlet

mass vanishes. In that limit, η is fully embedded in the third component of the triplets

Hu,dand one can easily see that all non-derivative couplings of η to the third generation

can be removed by rotations of the top and bottom quarks by the phase factors e±i˜ η/√2f.

After these rotations, η has only derivative couplings to the fermions, in other words, η is a

genuine Goldstone boson rather than a pGB. However, for a finite F, the singlet has f/F

suppressed components in Φu,dand the Yukawa couplings can no longer be rotated away.

The third generation quark loops then generate the operators like |HuΦd|2, which lead to η

acquiring a mass after electroweak symmetry breaking. The dominant contribution to the

singlet mass arises from loops of the bottom quark and its symmetry partners, and is given

by (for large tanβ)

m2

η≈3v2

EWy2

8π2

2

M2

F2

soft

?

log

?

y2

T+ M2

b1F2

2(M2

soft)

?

−

M2

M2

T

soft

log

?M2

T+ M2

M2

soft

T

?

+ 1

?

. (19)

For F ∼ 10 TeV and yb1 ∼ 0.1 this leads to η in the 1–3 GeV range.

contribution is subleading because it is not enhanced by logF. As we will discuss soon,

the above contribution is not enough to satisfy the existing LEP bounds if the Higgs mass

is in the 78–86 GeV window, in which case 6GeV<

a case, we can always invoke a small addition of non-collective couplings that enhance the

one-loop contributions to the η mass without spoiling naturalness. For example, we can add

non-collective Yukawa couplings in the bottom sector,

The top loop

∼ mη < 9.2 GeV is required. In such

˜ yb1HdQb1

c+ ˜ yb2ΦdQb2

c, (20)

with ˜ yb∼ 10−3. This leads to additional one-loop contributions to the η mass approximately

given by

m2

4π

η= cosβNc

F

f(yb1˜ yb1+ yb2˜ yb2)M2

softlogΛ

F.

(21)

In Fig. 3 we show a scatterplot of the Higgs and η masses by varying the input parameters

both without and with the non-collective bottom Yukawas.

Parameter scans

To illustrate the parameter space achievable in this model we have prepared two sets of

contour plots for f = 350 (Fig. 4) and 400 GeV (Fig. 5), where we show both the Higgs and

η masses and the fine tunings FT3,2. While these scans are not exhaustive, we can see that

large regions of the parameter space are open with Higgs masses in the 85−115 GeV range

9

Page 11

708090100

mh?GeV?

110120130140

0

1

2

3

4

mΗ?GeV?

f ? 350 GeV

7080 90100

mh?GeV?

110120130140

0

5

10

15

20

25

mΗ?GeV?

f ? 350 GeV

Figure 3: A scan of the parameter space for the achievable η and Higgs masses. In the left

panel we show the Higgs mass with ˜ yb1,b2= 0, and see that for the interesting range of Higgs

masses mη < 3 GeV. In the right we varied 0.001 < ˜ yb1 < 0.002. We can see that with

this non-collective Yukawa one can easily get mη in the 5–10 GeV range. We have fixed

f = 350,F =√2 · 104,Λ = 107GeV for both plots, and scanned the remaining parameters

in the regions 0.02 < y1< 0.3,1 < y2< 2.4,0.02 < yb1< 0.12 and 300 < Msoft< 1000 GeV.

and η masses between 3 and 9 GeV. The usual MSSM fine tuning FT2is mostly negligible,

while the UV completion dependent fine tuning FT3varies in the 3-10% range. For example,

for f = 350 GeV and a relatively small top Yukawa coupling y2≈ 1.64, the theory would

stays perturbative up to Λ ≈ 108TeV (which is also the Landau pole for the strong coupling

g3in the presence of one family of the vectorial states (V,Vc)), while the fine tuning FT3is

about 5% for Msoft∼ 300 GeV and a 90 GeV Higgs mass and 6 GeV η mass.

η decays: hiding the Higgs at LEP (and the LHC)

The last part of the plan to allow the Higgs to escape detection at LEP is to ensure that η

decays via a channel that is not well constrained by LEP. This is possible only if the singlet

is lighter than twice the b quark mass; if kinematically allowed, the η → b¯b decay channel

always dominates which excludes the Higgs lighter than 110 GeV. In the NMMSM context,

the dominant decay below 2b threshold is the decay into tau leptons. In our model this is

not the case. The reason is that the pGB singlet is embedded in the third component of

the Hu,dtriplets and it can couple to the SM quarks and leptons only via their mixing with

the heavy fermionic states. For τ, that mixing angle is suppressed by m2

heavy tau mass should be larger than few ×100 GeV. More precisely, the coupling of the

pGB singlet is

τ/M2

τ, where the

i˜ yτ(¯ τγ5τ)η˜ yτ≃

m3

τf

τv2

√2M2

EW

(22)

10

Page 12

3

4

5

115

6

7

8

9

10

11

12

80

85

90

95

100

105

110

120

300400500600700

1.4

1.6

1.8

2.0

2.2

ms?GeV?

y2

mhand mΗfor f?350 GeV

2?

3?

4?

4?

300

5?

6?

7?

8?

9?

10?

7?

10?

15?

20?

30?

50?

400 500600 700

1.4

1.6

1.8

2.0

2.2

ms?GeV?

y2

FT from doublet and triplet, f?350 GeV

Figure 4: On the left the contours of the Higgs mass (dashed red line) and the η mass

(solid black lines) as function of the universal soft breaking mass Msoftand the top Yukawa

y2. On the right, the necessary fine tunings FT3 (solid black) and FT2 (dashed red) in

percent. The kink in the contour lines at y2≈ 1.64 appears because the cut-off for larger

values is determined by the Landau pole of y2. These plots are based on the full numerical

1-loop Coleman-Weinberg potential, with f = 350 GeV, y1= 0.29,yb1= 0.1,yb2= 0, ˜ yb1=

0.001,tanβ = 10,F =√2 · 104and ˜ yb2= 0. The region in the lower left is excluded by the

LEP ξ2BR(h → bb) bound and in the lower right because mη> 2mb.

3

4

5

6

7

8

9

1011

12

80

300

85

90

95

100

105

110

115

120

400500600 700

1.4

1.6

1.8

2.0

2.2

ms?GeV?

y2

mhand mΗfor f?400 GeV

2?

3?

4?

4?

5?

6?

7?

8?

9?

10?

10?

15?

20?

30?

50?

300400500 600700

1.4

1.6

1.8

2.0

2.2

ms?GeV?

y2

FT from doublet and triplet, f?400 GeV

Figure 5: The same as in 4 for f = 400 GeV.

11

Page 13

Figure 6: The diagram for one loop η decay into gluons or photons.

where Mτ≈ yτ1F/√2. The corresponding decay width is

Γη→ττ≈

1

16π

?

1 − 4m2

τ/m2

η

mηm6

v4

τf2

EWM4

τ

.(23)

It depends on the sixth power of the tau mass and for this reason it is much more suppressed

than in the NMSSM models. For typical parameters, f ∼ 350 GeV and Mτ∼ 200 GeV, the

width into tau is in the 10−14− 10−13GeV range corresponding to order millimeter decay

length.

Because of the suppression of the ητ¯ τ coupling, the pGB singlet decays dominantly into

two gluons, via the loop diagram in Fig. 6 with bottom and top and their fermionic partners

running in the loop (the scalar partners do not contribute to this decay amplitude). Quite

generally, starting with the coupling i˜ yψη(ψγ5ψ) to light or heavy fermions, one-loop effects

generate the effective coupling [25]

κgηǫµνρσGa

µνGa

ρσ,κg=

g2

32π2

?

ψ

˜ yψ

mψc2(ψ)τψf(τψ)(24)

where

τψ= 4m2

ψ/m2

η

f(τ) =

?

arcsin2[τ−1/2]τ ≥ 1

τ < 1

−1

4

?log[(1 +√1 − τ)/(1 −√1 − τ)] − iπ?2

(25)

Furthermore, g = gs(mη) - the color SU(3) coupling at the scale of the singlet mass and c2

is the Dynkin index of the quark representation which is equal to 1/2 for the fundamental

representation. There is an analogous coupling κγto the photon field strengths with g → gem

and c2→ NcQ2

c− 1)|κg|2

π

ψ, Nc= 3. The decay width into two gluons and two photons is given by

Γη→gg= (N2

m3

η,Γη→γγ=|κγ|2

π

m3

η.(26)

The pGB singlet has the largest coupling to the bottom and the top quarks,

˜ yt≃

m3

tf

EWµ2

√2v2

V

,˜ yb≃

mbm2

√2v2

tf

EWµ2

V

.(27)

Since κg∼ ˜ yψ/mψfor mη≪ 2mψone would expect that the top and bottom loops dominate

the decay amplitude and give roughly the same contribution. This is not quite correct. In

12

Page 14

bb

ΓΓ

ΤΤ

cc

gg

246810

10?18

10?15

10?12

10?9

10?6

mΗ?GeV?

??GeV?

bb

gg

ΓΓ

ΤΤ

cc

246810

10?9

10?7

10?5

0.001

0.1

mΗ?GeV?

BR

Figure 7: The partial widths and the branching ratios of the pGB singlet η for decays into

gg, γγ, bb, ττ and cc. The parameters are f = 350 GeV, µV = 500 GeV, Mc= 400GeV ,

Mτ= 200 GeV.

the model at hand there is a sum rule?˜ yψ/mψ≈ O(1/F2) separately in the bottom, top,

and tau sectors. This sum rule is the consequence of the fact that η, at the leading order in

1/F, couples to a gauge symmetry current that is necessarily anomaly free (one can see in

the parametrization of eq. (5) that η couples to a combination of the SU(3) T8generator and

U(1)X). Thus, the operator ηG˜G cannot be generated by integrating out fermions; the lowest

allowed operator is ?ηG˜G. As a consequence, the amplitude is proportional to?˜ yψm2

which is non-vanishing and largely dominated by the SM bottom quark contribution. One

finds

κg≃

64π2

For the photons, one should replace g2

the decay amplitude has a practical consequence that the photonic branching ratio is more

suppressed than in the SM because |Qb| = |Qt|/2. At the end of the day the BR(η → γγ)

is of order 10−4. While this is not of much relevance for the LEP searches, the additional

suppression will make the LHC Higgs search even more difficult if possible at all.

The partial widths and the branching ratios of η are plotted in Fig. 7. One can see

that below the 2b threshold the dominant decay channel of the Higgs is the four-gluon

cascade decay. The branching ratio for h → ggγγ is of order 10−4. Discovering the Higgs

decaying almost exclusively to 4 gluons with such a small branching ratio into photons might

be impossible at the LHC [31]. Fermionic decay channels are also hugely suppressed, for

example the branching ratio for h → ggτ+τ−is in the range 10−5−10−3. This fermiophobic

feature of η implies that the recent D0 searches of h → ττµµ and h → 4µ [26,27], as well as

the BaBar and CLEO studies of υ decays [28,29] do not constrain the parameter space of

our model. Furthermore, we estimate the branching ratio for υ → γ +η to be of order 10−5,

which is safely below the CLEO limit of 10−4[30].

η/m3

ψ,

1

12√2

g2

s(mη)

m2

m2

η

b

emQ2

m2

µ2

tf

Vv2

EW

.(28)

s/2 → Ncg2

b. The bottom-loop domination of

13

Page 15

The best limits on the Higgs mass in our scenario follow from the analyses published by

the OPAL collaboration. Higgs masses smaller than ∼ 78 GeV are excluded by the decay-

mode independent search [32]. A search for h → 4j via a pseudoscalar has been performed

for the 78−86 GeV Higgs mass window in [33]. In our model, the Higgs branching fraction

into 4 jets is of order 80%, and moreover there is the suppression of 1 − v2

Higgs production cross section. This implies that already in the existing OPAL search the

η mass in the range 6GeV<

in the presence of small non-collective Yukawa couplings. The OPAL bound on the η-mass

can be understood qualitatively quite simply.5

highly boosted and both pairs of gluon jets will be collimated. The angle between the two

gluon jets is of order 4mη/mh, which for very small mηis too small for the four jets to be

independently resolved, in which case the very restrictive h → 2j exclusion limit will apply.

For Higgs masses above 86 GeV our scenario is not constrained by any existing experimental

analysis. Extrapolating the bound from [33] one expects that the bound on mηwill weaken

with increasing Higgs mass, and will dip below few GeV when mh>

EW/f2of the

∼mη

<

∼9.2GeV is allowed, which can be achieved in our model

When mη is very small, the etas will be

∼90 GeV.

Electroweak precision, unitarity and the radial mode

In our model, the pGB Higgs coupling to W and Z is suppressed by cos(˜ v/f) with respect

to the SM value. Because of that, the cancellation of logarithmic divergences in the gauge

boson self-energies is complete only after taking into account the radial modes ru,dassociated

with the oscillations of the VEVs of Hu,d. For large or moderate tanβ it is r = ruthat has

sizable couplings to W and Z: that coupling is suppressed by sin(˜ v/f) = vEW/f ∼ 1/2 with

respect to the SM Higgs coupling. The fact that there are two Higgs-like particle coupled

to the SM gauge bosons affects electroweak precision observables. As pointed out in [34],

since the electroweak S and T parameters depend logarithmically on the Higgs mass, one

can estimate the effect on S and T by defining an effective Higgs mass mEWPT which at

large tanβ is given by

?mr

mEWPT= mh

mh

?v2

EW/f2

, (29)

where m2

pendence is expected to be replaced everywhere by cos2(˜ v/f)logm2

For typical range of parameters mr∼ 300−400 GeV the corresponding mEWPTis in the safe

110−135 GeV range, so one expects the oblique corrections to be within the experimentally

preferred region. Another potential electroweak precision correction is the tree-level shift in

the Zb¯b vertex, which is due to the mixing of the physical left-handed bottom b with the

statesˆbQ(and also the right handed b with bV

of the left handed Zb¯b vertex is of the order

r≈ 4λHuf2is the mass of the radial mode r. This is because the SM logm2

h/Λ2de-

r/Λ2.

h/Λ2+sin2(˜ v/f)logm2

c). An explicit calculation shows that the shift

δg

g

∝ cos2βµ2

Vv2

B1m2

EW

m4

B2

?F(yb1˜ yb1+ yb2˜ yb2) + f cosβ(y2

b2+ ˜ y2

b1)?2,(30)

5We thank Paddy Fox for explaining this to us.

14

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