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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
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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
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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
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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
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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
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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)
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f ? 350GeV
f ? 400GeV
f ? 450GeV
80 85 9095
mh?GeV?
100105110
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)
100 120
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
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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)
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1.01.2 1.4 1.61.82.0
40
50
60
70
80
90
y2
mh
F ? 10TeV Msoft?700GeV y1? 0.5 yb1? 0.04
1.01.21.4 1.61.82.0
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
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70 8090 100
mh?GeV?
110 120130 140
0
1
2
3
4
mΗ?GeV?
f ? 350 GeV
70 8090 100
mh?GeV?
110120 130 140
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
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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?
400500600700
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?
300400500600700
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
2468 10
10?18
10?15
10?12
10?9
10?6
mΗ?GeV?
??GeV?
bb
gg
ΓΓ
ΤΤ
cc
2468 10
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
Page 16
where mB1,B2are the heavy bottom masses in the multi-hundred GeV range. We can see
that the shift is suppressed by cos2β ∼ 0.01, and has large additional mass suppression
factors reducing it well below the experimental bound of a few per mille. There are also
one-loop corrections to the Zb¯b vertex, and to the S and T parameters as well, coming for
instance from the extended Yukawa sector. While they deserve future investigations, a full
one-loop analysis is beyond the scope of the present paper.
The radial mode also takes part in unitarizing WW scattering. As usual in pseudo-
Goldstone Higgs theories, the cos(˜ v/f) suppression of the pGB Higgs coupling to the SM
gauge boson leads to an imperfect cancellation of the contributions that grow with E2in
the longitudinal gauge boson scattering amplitudes [35]. In our case however the theory is
weakly coupled as the radial mode completes the cancellation well below the unitarity bound
of ∼ 1 TeV. Thus we do not expect a significant rise in the WW scattering amplitude as
long as the mass of the radial mode is within the expected 300 − 400 GeV range.
The radial mode r itself might actually be more easily visible at the LHC than the Higgs
h. Since the radial mode has (suppressed) Higgs-like couplings and mass in the 300 − 400
GeV range, the standard heavy Higgs decay to WW and ZZ is allowed. In particular,
r → ZZ → 4l mode should be open and easily observable at the LHC despite the fact
that the production rate is suppressed by v2
may lead to the peculiar situation where the LHC discovers a “wrong” Higgs r which by
itself cannot account for either the electroweak precision fit or the unitarization of the WW
scattering, while the signal of the true Higgs is swamped by the QCD background. Apart
from the production cross section and couplings to gauge bosons, another phenomenological
difference with the standard Higgs would be a suppression of the h → tt branching ratio: the
symmetry protection of the Higgs potential typically results in a reduced Yukawa coupling
with the top for both the pGB Higgs and the radial mode.
EW/f2as compared to the SM Higgs. This
Conclusions
We have presented a simple complete model where the traditional tuning in the MSSM Higgs
potential is absent due to the pseudo-Goldstone nature of the Higgs. No additional sectors
have to be included to raise the Higgs quartic because the Higgs dominantly decays to the
fifth pseudo-Goldstone boson η, which can hide the Higgs at LEP. A tree-level coupling
responsible for the h → 2η is automatically present, and the mass of η is naturally in the
few to 10 GeV range. The leading Higgs decay turns out to be h → 2η → 4j, which is not
excluded by the existing LEP analyses for mh> 86 GeV, and for η masses in the 6–9.2 GeV
window it is also not excluded by the dedicated OPAL analyses in the 78 < mh< 86 GeV
window. While this Higgs is “buried” in the QCD background and would remain elusive
at the LHC, a heavy Higgs-like radial mode and many of the symmetry partners of the SM
fermions would be readily observable.
15
Page 17
Acknowledgments
We thank Jim Alexander, Roberto Contino, Paddy Fox, Bob McElrath, Maxim Perelstein,
Matt Reece, Gavin Salam, Jay Wacker, Lian-Tao Wang, Neal Weiner, Alvise Varagnolo,
Itay Yavin, and Dieter Zeppenfeld for useful discussions and comments. This research of
B.B. and C.C. has been supported in part by the NSF grant PHY-0757868. C.C. was also
supported in part by a U.S.-Israeli BSF grant. The work of A.F. was supported in part by
the Department of Energy grant DE-FG02-96ER40949.
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