arXiv:0704.1165v3  [hep-ph]  23 Sep 2007
April 2007
Electroweak Higgs as a pseudo-Goldstone boson
of broken scale invariance
Robert Foot, Archil Kobakhidze and Raymond R. Volkas 1
School of Physics, Research Centre for High Energy Physics,
The University of Melbourne, Victoria 3010, Australia
Abstract
We point out that it is possible to associate the electroweak Higgs boson with the pseudo-
Goldstone boson of broken scale invariance, thus resolving the hierarchy problem in a technically
natural way. We illustrate this idea with two specific gauge models. Besides being consistent with
all currently available experimental data, both models maintain the predictive power of the standard
model, since the first model has only one additional parameter beyond the standard model, and
the second has the same number of free parameters as the standard model.
1E-mail: foot@physics.unimelb.edu.au, archilk@physics.unimelb.edu.au, r.volkas@physics.unimelb.edu.au
http://arxiv.org/abs/0704.1165v3
1 Introduction
Understanding the origin of mass is one of the key problems within the standard model
(SM). The chiral nature of the gauge symmetry of the Standard Model forbids masses for
quarks and leptons, apart from right-handed neutrino Majorana masses. Taking neutrinos
to be Dirac, the only mass parameter in the theory is the electroweak µ2 mass parameter
in the Higgs potential. Setting it to zero renders the SM scale invariant at the classical
level. However, the scale invariance is known to be anomalous; it is broken at the quantum
level. This manifests through the important effect called dimensional transmutation. In
particular, in a scale invariant theory a scalar field can develop a vacuum expectation value
(VEV) radiatively as a result of the quantum conformal anomaly [1]. On general grounds
one can argue that the existence of a light scalar field, with its mass generated entirely at
loop level, is inevitable within scale invariant models that involve also multiple scalar fields
[2]. This light scalar field is nothing but a pseudo-Goldstone boson (PGB) accompanying
the breaking of the anomalous scale invariance. It is tempting to associate this PGB with
the electroweak Higgs boson. The reasons are obvious. Firstly, a massless theory is more
predictive than the corresponding theory with a priori unknown mass parameters. Secondly,
and perhaps more importantly, the electroweak scale generated through the dimensional
transmutation will be radiatively stable, thus resolving the technical aspect of the hierarchy
problem. Unfortunately, this very appealing theoretical framework applied within the SM
predicts a very light Higgs boson with mh < 10 GeV and also requires the top quark to be
light: mt
<∼ 40 GeV . Both of these predictions are in sharp contradiction with observations.
A quick inspection of the Coleman-Weinberg effective potential reveals the way to circum-
vent the above problem in weakly-coupled (perturbative) theories. One needs to ensure the
overall dominance of the bosonic contributions to the effective potential over the fermionic
ones, such that the experimental lower bound on the Higgs boson mass is satisfied while
keeping the coupling constant in the perturbative domain. The simplest way to achieve
this is by simply adding one (or more) scalar fields to the theory.2 The minimal model of
this type involves a Higgs potential with just three parameters – which is only one more
parameter than the SM case. We study this model in section II, where we show that it does
provide a phenomenological consistent theoretical framework to realise the Higgs boson as a
PGB of broken scale invariance.
Extended scalar sectors are a feature of many theories beyond the standard model. One
2Scale invariant models with additional scalars have been considered in Refs. [4] and [5] (see also the
discussion of the scale-invariant case in Ref. [6]). The philosophy of those models was somewhat different
though, with the additional scalar gaining a large VEV in both models which means that the electroweak
Higgs could not be interpreted as the PGB of broken scale invariance. Also in Ref. [4] there was an additional
U(1)X gauge boson. The models which we consider are simpler, with less parameters than these alternative
models.
such theory, with a particularly simple Higgs sector, is the mirror model [3]. In that theory,
one essentially has two isomorphic sectors of particles, the standard particles and a ‘mirror’
sector. The mirror particles are governed by a Lagrangian of exactly the same form as the
standard model, so that a discrete Z2 mirror symmetry can be defined interchanging the
ordinary and the mirror particles. If we make the theory scale invariant, by eliminating
the µ2 mass parameter in the Higgs potential, then we can generate electroweak symmetry
breaking radiatively, via the Coleman-Weinberg mechanism. The Higgs potential then has
just two parameters (the same as in the SM) since the Z2 mirror symmetry in the model
fixes the quartic coupling constant in the mirror sector to be the same as the corresponding
coupling constant in the ordinary particle sector. We show in section III that this classically
scale invariant model is phenomenologically consistent, and also realises the Higgs boson as
a PGB of spontaneously broken scale invariance.
2 The next-to-minimal scale-invariant Standard Model
Consider a minimal extension of the scale-invariant SM with a single extra real scalar field.
The most general renormalisable potential is
V0(φ, S) =
(φ†φ)2 +
(φ†φ)S2 , (1)
where φ is the electroweak Higgs doublet and S is the real singlet field. observe that the above
potential (as well as total Lagrangian) has an accidental discrete Z2 symmetry: S → −S.
We parameterise the fields (in the unitary gauge) as:
, S = r sinω (2)
In this parameterisation, the potential (1) takes the form
V0(r, ω) = r
cos4 ω +
sin4 ω +
sin2 ω cos2 ω
. (3)
Note that the radial component r of the Higgs fields (2) factors out in the absence of the
tree-level mass parameter. The renormalised quantum-corrected potential, in addition to
(3), includes a sum of δVk−loop contributions generated at k-loop level (k = 1, 2, . . .). Each
k-loop contribution is a kth order polynomial in log(φ/Λ), where Λ is some renormalisation
scale. Perturbation theory is valid if the logarithms are not too large, so that V0 > δV1−loop >
δV2−loop > . . . is satisfied. In general, the minimisation of even a 1-loop corrected effective
potential involving multiple scalars cannot be done analytically, but one may resort to a
numerical analysis. Here we instead follow the approximate analytic method suggested in
[2] which is suitable in weakly coupled scale invariant theories.
Following [2], we first ignore perturbatively small radiative corrections and concentrate on
the tree-level potential (3). Minimising the potential, assuming that r 6= 0 but is otherwise
at this stage arbitrary, leads to two possible cases (we ignore a third case with unbroken
electroweak symmetry because it is phenomenologically not viable). If λ3 > 0 then
〈sinω〉 = 0 , 〈r〉 =
2〈φ〉 ≡ v ≈ 246 GeV , < S >= 0 , (4)
λ1(Λ) = 0 . (5)
In this case only the electroweak Higgs develops a nonzero VEV. If λ3 < 0 then
〈tan2 ω〉 = ǫ ,
2〈φ〉 = 〈r〉
1 + ǫ
≡ v ≈ 246 GeV , 〈S〉 = v〈tanω〉 , (6)
λ3(Λ) +
λ1(Λ)λ2(Λ) = 0 (7)
where ǫ ≡
λ1(Λ)
λ2(Λ)
. In this case both scalar fields develop VEVs and the discrete Z2 symmetry
is also broken spontaneously.3
Relations such as Eqs.(5) and (7) can be satisfied by an appropriate choice of the renor-
malizsation point µ = Λ [2], where the running coupling constants depend on µ, and Λ is the
specific value where the required relations hold. In each case, the tree-level potential then
has a flat direction along the vacuum solution, and the relation Eq.(5) [or Eq.(7)] removes a
dimensionless parameter in favour of the renormalisation point Λ which has the dimension of
mass (dimensional transmutation). Because the tree-level potential vanishes along a specific
direction, the 1-loop correction to it will dominate along that direction.
Next we calculate the tree-level masses by expanding the Higgs potential, Eq.(1), around
the vacuum: φ = 〈φ〉+φ′, S = 〈S〉+S ′. In each case there are two physical scalars, but only
one of these gains mass at tree-level, since there is a flat direction in the Higgs potential.
Let us call H the state that gets mass at tree-level, and h the state which is massless at
tree-level (the PGB of broken scale invariance). For the first case, where λ3 > 0 ⇒ 〈S〉 = 0
[Eq.(4)], we find
m2H =
, H = S. (8)
3This might cause a problem with cosmological domain walls unless 〈S〉 is sufficiently small. If it is very
small but nonzero, then a network of domain walls would form with an equation of state (p = wρ) parameter
w = −2/3 in the non-relativistic limit. This is somewhat interesting for dark energy reasons, though present
indications are that a w value closer to −1 is preferred by the data.
The PGB in this case is h = φ′0.
In the second case, where λ3 < 0 and both S and φ gain VEVs, we find
m2H = λ1v
2 − λ3v2 , H = − sinωφ′0 + cosωS ′. (9)
In this case the PGB is h = cosωφ′0 + sinωS
Let us now calculate the mass of the PGB boson for each pattern of symmetry breaking.
The 1-loop correction to the tree-level potential (3) (along the radial direction) has the form
δV1−loop = Ar
4 + Br4 log
, (10)
where
64π2〈r〉4
M4V log
M4S log
− 4Tr
M4F log
64π2〈r〉4
3TrM4V + TrM
S − 4TrM4F
. (12)
The traces in the above equations go over all internal degrees of freedom, and MV,S,F are the
tree-level masses respectively for vectors, scalars and fermions evaluated for the given VEV
pattern.
The stationary condition
∂δV1−loop
|r=〈r〉 = 0 implies the relation
. (13)
The PGB mass can be calculated directly from δV1−loop. Using (13) one finds [2]:
m2h =
∂2δV1−loop
|r=〈r〉 = 8B〈r〉2
8π2〈r〉2
3TrM4V + TrM
S − 4TrM4F
. (14)
Applying this equation to determine the mass of the PGB, we find:
m2h =
8π2〈r〉2
6m4W + 3m
H − 12m4t
m4H cos
8π2v2
, (15)
since we need mH to dominate over the other terms if the PGB mass is to be larger than
the experimental lower limit of about ∼ 115 GeV.
Precision electroweak tests put an upper bound on the Higgs boson mass. The current
upper limit for the standard model Higgs is mhiggs < MEW , with MEW ≈ 186 GeV at 95%
C.L. [7]. Since the Higgs boson mass gives a radiative correction at l-loop level via a log
term, we can find the corresponding limit in this model by the replacement λ2SM logm
higgs →
λ2SM cos
2 ω logm2h + λ
SM sin
2 ω logm2H . Thus, the limit on the scale invariant model from
precision electroweak tests is
mH < MEW , (16)
where cω ≡ cosω. Clearly, in the first symmetry breaking scenario, where cosω = 1, this
bound only constrains the mass of the PGB h to be less than MEW which can easily be
satisfied (as is also the case in the SM). In the second case, where cosω is essentially a free
parameter, the above bound gives a contraint on ω (given the relation, Eq.15). Using the
current experimental bound, MEW ≈ 186 GeV, we find that tanω < 0.65. That is to say,
the PGB mainly “resides” in the electroweak doublet.
Note that the above analysis with one real scalar field can be simply extended to N real
scalar fields. Taking for simplicity an O(N) symmetric potential, we simply need to replace
S2 → ∑Ni=1 S2i in the potential, Eq.(1). In this case only the λ3 > 0 region, where 〈Si〉 = 0,
is phenomenologically viable, since having 〈Si〉 6= 0 would lead to massless Goldstone bosons
from the spontaneous breaking of O(N) symmetry. Also, if 〈Si〉 = 0 we can also give S
gauge quantum numbers. For example, having S complex and transforming as an SU(3)c
colour triplet would be equivalent to having N = 6 real scalar fields with an O(6) symmetric
potential. Having N scalar fields, will give a factor N in the right-hand side of Eq.(15) and
thus reduce the mass of the heavy scalar (for a fixed mh).
To find the domain of validity of perturbation theory we need to look at the renormali-
sation group equations [8]:
= 3λ21 +
N + 8
λ22 + λ
3λ1λ3
(N + 2)
λ2λ3 +
, (17)
where t = log µ. Contributions from gauge and Yukawa coupling constants can be approxi-
mately neglected relative to the Higgs potential couplings constants.
These equations must be supplemented by the constraint Eq.(5) or (7) for the two sym-
metry breaking scenarios of interest. Due to the relatively large mH mass, the position of
the Landau pole, µL, is typically only a few orders of magnitude above the weak scale in
both symmetry breaking scenarios. For example, for mh at the experimental limit of 115
GeV, taking N = 1 (N = 6) and the case where λ3 > 0 so that 〈S〉 = 0, we find, numerically
solving the equations, that µL ≈ 20Λ ≈ 104 GeV (µL ≈ 4× 102Λ ≈ 105 GeV). [Note that Λ
is determined from Eq.(13)].
3 Electroweak Higgs as a PGB of broken scale invari-
ance in mirror models
In the previous section we showed that extending the Higgs sector in a scale invariant theory
allows consistent models giving a naturally light and radiatively stable Higgs boson. Many
theories beyond the SM actually require an extended Higgs sector. One theory with a
particularly simple Higgs potential is the mirror matter model [3]. In the simplest version
of that theory each type of ordinary particle (other than the graviton) has a distinct mirror
partner. The ordinary and mirror particles form parallel sectors, each with gauge symmetry
GSM = SU(3)⊗SU(2)⊗U(1), so that the overall gauge group is GSM⊗GSM. The interactions
within each sector are governed by Lagrangians of exactly the same form, except that mirror
weak interactions are right-handed.4 In other words, the full Lagrangian has the form
L = L1 + L2 + Lmix , (18)
where L1 is the usual Lagrangian of the SM, while L2 is the Lagrangian for the mirror SM.
They are related by a parity symmetry P, such that PL1P−1 = L2, which is imposed as
an exact symmetry of the whole Lagrangian: PLP−1 = L. The third term Lmix contains
only two parity invariant renormalisable interactions that couple the ordinary sector with
the mirror one:
Lmix = ǫF 1µνF 2 µν + 2λ(φ
1φ1)(φ
2φ2) , (19)
where F 1µν and F
µν are the U(1) field strengths for the ordinary and mirror sectors and φ1
and φ2 are ordinary and mirror electroweak Higgs doublets, respectively. Thus we have only
two extra parameters, ǫ and λ, in addition to those of the ordinary SM. The U(1) kinetic
mixing term will not be of interest to us here.
The most general tree-level Higgs potential of the scale invariant mirror model can be
expressed as
V0(φi) = λ
1φ1 + φ
1φ1 + φ
. (20)
This potential is bounded from below if λ + δ ≥ 0 and λ + δ
≥ 0. We take the Higgs field
φi of each sector in unitary gauge and express them as
, φ2 =
. (21)
4There is a related model where the interchange symmetry is just an internal Z2, unrelated to parity. In
that version, the “shadow” weak interactions are left-handed.
Then the potential (20) takes the form,
V0(r, ω) =
λ+ δ(cos4 ω + sin4 ω)
. (22)
As in the previous section, we can minimise the potential, assuming r 6= 0, which leads to
two cases (depending on the sign of λ):
Broken − P case for λ > 0 : sinω = 0 or cosω = 0 requiring λ+ δ = 0;
Unbroken − P case for λ < 0 : sinω = cosω = 1√
requiring λ+
= 0. (23)
As before, the relations between λ and δ will be satisfied by an appropriate choice of the
renormalisation point µ = Λ. For the broken-P case, we have either 〈φ1〉 or 〈φ2〉 being zero.
Since we are identifying sector 1 as the ordinary sector, we shall concentrate on the sinω = 0
configuration.
Next we calculate tree-level mass-squared matrix at µ = Λ:
Broken− P Case : M2ij = λ〈r〉2
Unbroken − P case : M2ij = λ〈r〉2
The above mass matrices each have vanishing determinant, so one of the physical scalar
fields, the PGB, is massless at tree level. As usual, the conformal anomaly means it obtains
a relatively small mass from the 1-loop correction to V0.
Also, it can be easily seen that the matrices in Eq.(24) have non-negative eigenvalues
provided that λ > 0 (λ < 0) for the broken- (unbroken-) P cases. Thus, there are only two
physically distinct minima of the potential (22):
〈r〉 ≡ v ≈ 246 GeV, 〈cosω〉 = 1, if λ > 0 , (25)
〈r〉 ≡
2 · 246 GeV, 〈cosω〉 =
, if λ < 0 . (26)
In case (25) the electroweak symmetry is broken in the SM sector, while the electroweak
symmetry in the mirror sectors is intact5, and hence the Z2 parity symmetry is spontaneously
broken. In this model, the scalar sector consists of the standard Higgs boson, which is
massless at tree level, and a complex mirror-doublet of bosons with mass squared: m2H =
λv2 ≈ λ(246)2 GeV2.
5SU(2) ⊗ U(1) electroweak symmetry in the mirror sector is eventually broken through mirror SU(3)
quark condensation. For details, see Ref. [9].
In case (26), the SU(2)×U(1) symmetry is broken in each sector so the Z2 mirror sym-
metry remains exact. Consequently, the gauge bosons and fermions in each sector obtain
the same masses as the corresponding particles in the SM. At tree level there is one mas-
sive scalar, with m2H = −4λv2 ≈ −4λ(246) GeV2, and one massless state. This massless
state corresponds to the PGB of broken scale invariance. The mass eigenstates are parity
eigenstates, maximal superpositions of the ordinary and mirror physical Higgs bosons.
Let us now calculate the mass of the PGB boson for each pattern of symmetry breaking.
As briefly reviewed in section II, the mass of the PGB is, in general, given by Eq.(14).
Applying this to the spontaneously broken mirror symmetry case (25) we obtain,
m2h ≃
8π2v2
6m4W + 3m
Z + 4m
H − 12m4t
2π2v2
for mh
>∼ 115 GeV, (27)
while for the unbroken mirror symmetry case (26) we find:
m2h ≃
16π2v2
12m4W + 6m
H − 24m4t
16π2v2
for mh
>∼ 115 GeV. (28)
Thus, we effectively have a mass relation between the light and heavy Higgs bosons in the
models. Numerically, we obtain the approximate relations:
115 GeV
360 GeV Broken case (Eq.25);
115 GeV
600 GeV Unbroken case (Eq.26). (29)
Phenomenologically, the broken mirror symmetry case, Eq.(25), mimics the SM. The
light Higgs, h, couples in exactly the same way as does the SM Higgs field, while the heavier
states, H , couple to the mirror sector. In this case, the experimental lower limit on mh is
the same as the SM limit of approximately 115 GeV. Also, the upper bound on mh inferred
from precision electroweak measurements is mh < MEW ≈ 186 [7]. There is no difficulty in
satisfying these constraints.
In the case of the unbroken mirror symmetry, Eq.(26), the light Higgs field h, couples
to both of the sectors, with coupling strength 1/
2 compared with the SM Higgs. Thus
the limit from precision electroweak measurements in this model is given by Eq.(16) with
c2ω = 1/2. However, for MEW ≈ 186 GeV, this bound is inconsistent with the relation,
Eq.(28). We conclude that the Coleman-Weinberg mechanism is consistent with existing
phenomenological bounds only for the broken-P situation6. Below, we therefore discuss the
broken-P case only.
The experimental lower bound on the mass of the SM Higgs boson, mh > 115 GeV, can
be translated into a lower bound on the coupling λ2(Λ) ≡ 2λ(Λ):
λ2(Λ) > 4.2
115 GeV
, (30)
for the broken P case of Eq.(25). To find the domain of validity of perturbation theory we
look, as before, at renormalisation group equations:
= 3λ21 + λ
2 , (31)
= 3λ1λ2 +
λ22 , (32)
where 1
λ1 = λ + δ. As λ2(Λ) is significantly larger than any other coupling in the SM we
can approximately neglect top-quark and gauge boson contributions. These equations must
be supplemented by the constraint equation (23), which reads
λ1(Λ) = 0 , (33)
for the case (25). Numerically solving these equations, we find that the position, µL, of the
Landau pole is µL ≈ 3 × 102Λ ≈ 105 GeV (for mh at the experimental limit of 115 GeV).
The perturbative domain thus extends confortably above the electroweak scale.
4 Conclusion
We have examined the idea that the standard model Higgs boson might be the pseudo-
Goldstone boson of broken scale invariance. The simplest version of this idea is the original
Coleman-Weinberg model. The problem there is that the mass of the Higgs is too small,
and the spontaneous symmetry breaking can only occur if the top quark is also very light.
However, if there is an extended Higgs sector, then the additional scalar degrees of freedom
can compensate for the heavy top quark, and give a Higgs mass in excess of the experimental
lower limit, currently around 115 GeV.
Specifically, we have considered two phenomenologically consistent models. The first
involves the addition of one (or more) real scalar fields. The additional bosonic degrees
of freedom can lead to phenomenologically successful electroweak symmetry breaking. An
6Of course this conclusion is only for the minimal mirror model with two sectors. In this context it will
be interesting to study the case of generalized mirror models with N-sectors [10]. However, we will leave this
study for the future.
even more constrained symmetry breaking sector arises in the mirror model, which has a
discrete symmetry interchanging the standard model Higgs with a mirror partner. The
discrete symmetry eliminates one parameter in the Higgs potential, and leads to a consistent
electroweak symmetry breaking with the same number of parameters as in the standard
model case.
The proposed scale-invariant models (and their generalizations) can be testable in up-
coming LHC experiments in the case of non-zero cosω. The light PGB Higgs interacts with
Standard Model particles with couplings reduced by factor cosω relative to the correspond-
ing couplings of the Standard Model Higgs boson. In addition heavier Higgs-like boson with
a mass correlated with the mass of PGB Higgs (see Eq.(15) can be observed. The couplings
of this heavy boson with Standard model particles is suppressed by sinω and its total decay
width will be dominated by the decay width into two PGB Higgses. In the case of cosω = 0,
the difference between the PGB Higgs and the Standard Model Higgs is rooted in the scalar
potential, and, most probably, LHC will not be capable to distinguish among them. Future
linear collider (e.g. the proposed ILC) should be able to study this case.
5 Acknowledgements
This work was supported by the Australian Research Council.
References
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[2] E. Gildener and S. Weinberg, Phys. Rev. D 13 (1976) 3333.
[3] R. Foot, H. Lew and R. R. Volkas, Phys. Lett. B 272 (1991) 67;
The concept of mirror matter was discussed prior to the advent of the standard model
in T. D. Lee and C. N. Yang, Phys. Rev. 104 (1956) 256; I. Kobzarev, L. Okun
and I. Pomeranchuk, Sov. J. Nucl. Phys. 3 (1996) 837; M. Pavsic, Int. J. Theor. Phys.
9 (1974) 229.
[4] R. Hempfling, Phys. Lett. B 379 (1996) 153 [arXiv:hep-ph/9604278];
W. F. Chang, J. N. Ng and J. M. S. Wu, arXiv:hep-ph/0701254.
[5] K. A. Meissner and H. Nicolai, arXiv:hep-th/0612165.
[6] J. R. Espinosa and M. Quiros, arXiv:hep-ph/0701145.
[7] W. M. Yao et al. [Particle Data Group], J. Phys. G 33 (2006) 1.
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[9] R. Foot and H. Lew, arXiv:hep-ph/9411390.
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  We point out that it is possible to associate the electroweak Higgs boson
with the pseudo-Goldstone boson of broken scale invariance, thus resolving the
hierarchy problem in a technically natural way. We illustrate this idea with
two specific gauge models. Besides being consistent with all currently
available experimental data, both models maintain the predictive power of the
standard model, since the first model has only one additional parameter beyond
the standard model, and the second has the same number of free parameters as
the standard model.

Introduction
Understanding the origin of mass is one of the key problems within the standard model
(SM). The chiral nature of the gauge symmetry of the Standard Model forbids masses for
quarks and leptons, apart from right-handed neutrino Majorana masses. Taking neutrinos
to be Dirac, the only mass parameter in the theory is the electroweak µ2 mass parameter
in the Higgs potential. Setting it to zero renders the SM scale invariant at the classical
level. However, the scale invariance is known to be anomalous; it is broken at the quantum
level. This manifests through the important effect called dimensional transmutation. In
particular, in a scale invariant theory a scalar field can develop a vacuum expectation value
(VEV) radiatively as a result of the quantum conformal anomaly [1]. On general grounds
one can argue that the existence of a light scalar field, with its mass generated entirely at
loop level, is inevitable within scale invariant models that involve also multiple scalar fields
[2]. This light scalar field is nothing but a pseudo-Goldstone boson (PGB) accompanying
the breaking of the anomalous scale invariance. It is tempting to associate this PGB with
the electroweak Higgs boson. The reasons are obvious. Firstly, a massless theory is more
predictive than the corresponding theory with a priori unknown mass parameters. Secondly,
and perhaps more importantly, the electroweak scale generated through the dimensional
transmutation will be radiatively stable, thus resolving the technical aspect of the hierarchy
problem. Unfortunately, this very appealing theoretical framework applied within the SM
predicts a very light Higgs boson with mh < 10 GeV and also requires the top quark to be
light: mt
<∼ 40 GeV . Both of these predictions are in sharp contradiction with observations.
A quick inspection of the Coleman-Weinberg effective potential reveals the way to circum-
vent the above problem in weakly-coupled (perturbative) theories. One needs to ensure the
overall dominance of the bosonic contributions to the effective potential over the fermionic
ones, such that the experimental lower bound on the Higgs boson mass is satisfied while
keeping the coupling constant in the perturbative domain. The simplest way to achieve
this is by simply adding one (or more) scalar fields to the theory.2 The minimal model of
this type involves a Higgs potential with just three parameters – which is only one more
parameter than the SM case. We study this model in section II, where we show that it does
provide a phenomenological consistent theoretical framework to realise the Higgs boson as a
PGB of broken scale invariance.
Extended scalar sectors are a feature of many theories beyond the standard model. One
2Scale invariant models with additional scalars have been considered in Refs. [4] and [5] (see also the
discussion of the scale-invariant case in Ref. [6]). The philosophy of those models was somewhat different
though, with the additional scalar gaining a large VEV in both models which means that the electroweak
Higgs could not be interpreted as the PGB of broken scale invariance. Also in Ref. [4] there was an additional
U(1)X gauge boson. The models which we consider are simpler, with less parameters than these alternative
models.
such theory, with a particularly simple Higgs sector, is the mirror model [3]. In that theory,
one essentially has two isomorphic sectors of particles, the standard particles and a ‘mirror’
sector. The mirror particles are governed by a Lagrangian of exactly the same form as the
standard model, so that a discrete Z2 mirror symmetry can be defined interchanging the
ordinary and the mirror particles. If we make the theory scale invariant, by eliminating
the µ2 mass parameter in the Higgs potential, then we can generate electroweak symmetry
breaking radiatively, via the Coleman-Weinberg mechanism. The Higgs potential then has
just two parameters (the same as in the SM) since the Z2 mirror symmetry in the model
fixes the quartic coupling constant in the mirror sector to be the same as the corresponding
coupling constant in the ordinary particle sector. We show in section III that this classically
scale invariant model is phenomenologically consistent, and also realises the Higgs boson as
a PGB of spontaneously broken scale invariance.
2 The next-to-minimal scale-invariant Standard Model
Consider a minimal extension of the scale-invariant SM with a single extra real scalar field.
The most general renormalisable potential is
V0(φ, S) =
(φ†φ)2 +
(φ†φ)S2 , (1)
where φ is the electroweak Higgs doublet and S is the real singlet field. observe that the above
potential (as well as total Lagrangian) has an accidental discrete Z2 symmetry: S → −S.
We parameterise the fields (in the unitary gauge) as:
, S = r sinω (2)
In this parameterisation, the potential (1) takes the form
V0(r, ω) = r
cos4 ω +
sin4 ω +
sin2 ω cos2 ω
. (3)
Note that the radial component r of the Higgs fields (2) factors out in the absence of the
tree-level mass parameter. The renormalised quantum-corrected potential, in addition to
(3), includes a sum of δVk−loop contributions generated at k-loop level (k = 1, 2, . . .). Each
k-loop contribution is a kth order polynomial in log(φ/Λ), where Λ is some renormalisation
scale. Perturbation theory is valid if the logarithms are not too large, so that V0 > δV1−loop >
δV2−loop > . . . is satisfied. In general, the minimisation of even a 1-loop corrected effective
potential involving multiple scalars cannot be done analytically, but one may resort to a
numerical analysis. Here we instead follow the approximate analytic method suggested in
[2] which is suitable in weakly coupled scale invariant theories.
Following [2], we first ignore perturbatively small radiative corrections and concentrate on
the tree-level potential (3). Minimising the potential, assuming that r 6= 0 but is otherwise
at this stage arbitrary, leads to two possible cases (we ignore a third case with unbroken
electroweak symmetry because it is phenomenologically not viable). If λ3 > 0 then
〈sinω〉 = 0 , 〈r〉 =
2〈φ〉 ≡ v ≈ 246 GeV , < S >= 0 , (4)
λ1(Λ) = 0 . (5)
In this case only the electroweak Higgs develops a nonzero VEV. If λ3 < 0 then
〈tan2 ω〉 = ǫ ,
2〈φ〉 = 〈r〉
1 + ǫ
≡ v ≈ 246 GeV , 〈S〉 = v〈tanω〉 , (6)
λ3(Λ) +
λ1(Λ)λ2(Λ) = 0 (7)
where ǫ ≡
λ1(Λ)
λ2(Λ)
. In this case both scalar fields develop VEVs and the discrete Z2 symmetry
is also broken spontaneously.3
Relations such as Eqs.(5) and (7) can be satisfied by an appropriate choice of the renor-
malizsation point µ = Λ [2], where the running coupling constants depend on µ, and Λ is the
specific value where the required relations hold. In each case, the tree-level potential then
has a flat direction along the vacuum solution, and the relation Eq.(5) [or Eq.(7)] removes a
dimensionless parameter in favour of the renormalisation point Λ which has the dimension of
mass (dimensional transmutation). Because the tree-level potential vanishes along a specific
direction, the 1-loop correction to it will dominate along that direction.
Next we calculate the tree-level masses by expanding the Higgs potential, Eq.(1), around
the vacuum: φ = 〈φ〉+φ′, S = 〈S〉+S ′. In each case there are two physical scalars, but only
one of these gains mass at tree-level, since there is a flat direction in the Higgs potential.
Let us call H the state that gets mass at tree-level, and h the state which is massless at
tree-level (the PGB of broken scale invariance). For the first case, where λ3 > 0 ⇒ 〈S〉 = 0
[Eq.(4)], we find
m2H =
, H = S. (8)
3This might cause a problem with cosmological domain walls unless 〈S〉 is sufficiently small. If it is very
small but nonzero, then a network of domain walls would form with an equation of state (p = wρ) parameter
w = −2/3 in the non-relativistic limit. This is somewhat interesting for dark energy reasons, though present
indications are that a w value closer to −1 is preferred by the data.
The PGB in this case is h = φ′0.
In the second case, where λ3 < 0 and both S and φ gain VEVs, we find
m2H = λ1v
2 − λ3v2 , H = − sinωφ′0 + cosωS ′. (9)
In this case the PGB is h = cosωφ′0 + sinωS
Let us now calculate the mass of the PGB boson for each pattern of symmetry breaking.
The 1-loop correction to the tree-level potential (3) (along the radial direction) has the form
δV1−loop = Ar
4 + Br4 log
, (10)
where
64π2〈r〉4
M4V log
M4S log
− 4Tr
M4F log
64π2〈r〉4
3TrM4V + TrM
S − 4TrM4F
. (12)
The traces in the above equations go over all internal degrees of freedom, and MV,S,F are the
tree-level masses respectively for vectors, scalars and fermions evaluated for the given VEV
pattern.
The stationary condition
∂δV1−loop
|r=〈r〉 = 0 implies the relation
. (13)
The PGB mass can be calculated directly from δV1−loop. Using (13) one finds [2]:
m2h =
∂2δV1−loop
|r=〈r〉 = 8B〈r〉2
8π2〈r〉2
3TrM4V + TrM
S − 4TrM4F
. (14)
Applying this equation to determine the mass of the PGB, we find:
m2h =
8π2〈r〉2
6m4W + 3m
H − 12m4t
m4H cos
8π2v2
, (15)
since we need mH to dominate over the other terms if the PGB mass is to be larger than
the experimental lower limit of about ∼ 115 GeV.
Precision electroweak tests put an upper bound on the Higgs boson mass. The current
upper limit for the standard model Higgs is mhiggs < MEW , with MEW ≈ 186 GeV at 95%
C.L. [7]. Since the Higgs boson mass gives a radiative correction at l-loop level via a log
term, we can find the corresponding limit in this model by the replacement λ2SM logm
higgs →
λ2SM cos
2 ω logm2h + λ
SM sin
2 ω logm2H . Thus, the limit on the scale invariant model from
precision electroweak tests is
mH < MEW , (16)
where cω ≡ cosω. Clearly, in the first symmetry breaking scenario, where cosω = 1, this
bound only constrains the mass of the PGB h to be less than MEW which can easily be
satisfied (as is also the case in the SM). In the second case, where cosω is essentially a free
parameter, the above bound gives a contraint on ω (given the relation, Eq.15). Using the
current experimental bound, MEW ≈ 186 GeV, we find that tanω < 0.65. That is to say,
the PGB mainly “resides” in the electroweak doublet.
Note that the above analysis with one real scalar field can be simply extended to N real
scalar fields. Taking for simplicity an O(N) symmetric potential, we simply need to replace
S2 → ∑Ni=1 S2i in the potential, Eq.(1). In this case only the λ3 > 0 region, where 〈Si〉 = 0,
is phenomenologically viable, since having 〈Si〉 6= 0 would lead to massless Goldstone bosons
from the spontaneous breaking of O(N) symmetry. Also, if 〈Si〉 = 0 we can also give S
gauge quantum numbers. For example, having S complex and transforming as an SU(3)c
colour triplet would be equivalent to having N = 6 real scalar fields with an O(6) symmetric
potential. Having N scalar fields, will give a factor N in the right-hand side of Eq.(15) and
thus reduce the mass of the heavy scalar (for a fixed mh).
To find the domain of validity of perturbation theory we need to look at the renormali-
sation group equations [8]:
= 3λ21 +
N + 8
λ22 + λ
3λ1λ3
(N + 2)
λ2λ3 +
, (17)
where t = log µ. Contributions from gauge and Yukawa coupling constants can be approxi-
mately neglected relative to the Higgs potential couplings constants.
These equations must be supplemented by the constraint Eq.(5) or (7) for the two sym-
metry breaking scenarios of interest. Due to the relatively large mH mass, the position of
the Landau pole, µL, is typically only a few orders of magnitude above the weak scale in
both symmetry breaking scenarios. For example, for mh at the experimental limit of 115
GeV, taking N = 1 (N = 6) and the case where λ3 > 0 so that 〈S〉 = 0, we find, numerically
solving the equations, that µL ≈ 20Λ ≈ 104 GeV (µL ≈ 4× 102Λ ≈ 105 GeV). [Note that Λ
is determined from Eq.(13)].
3 Electroweak Higgs as a PGB of broken scale invari-
ance in mirror models
In the previous section we showed that extending the Higgs sector in a scale invariant theory
allows consistent models giving a naturally light and radiatively stable Higgs boson. Many
theories beyond the SM actually require an extended Higgs sector. One theory with a
particularly simple Higgs potential is the mirror matter model [3]. In the simplest version
of that theory each type of ordinary particle (other than the graviton) has a distinct mirror
partner. The ordinary and mirror particles form parallel sectors, each with gauge symmetry
GSM = SU(3)⊗SU(2)⊗U(1), so that the overall gauge group is GSM⊗GSM. The interactions
within each sector are governed by Lagrangians of exactly the same form, except that mirror
weak interactions are right-handed.4 In other words, the full Lagrangian has the form
L = L1 + L2 + Lmix , (18)
where L1 is the usual Lagrangian of the SM, while L2 is the Lagrangian for the mirror SM.
They are related by a parity symmetry P, such that PL1P−1 = L2, which is imposed as
an exact symmetry of the whole Lagrangian: PLP−1 = L. The third term Lmix contains
only two parity invariant renormalisable interactions that couple the ordinary sector with
the mirror one:
Lmix = ǫF 1µνF 2 µν + 2λ(φ
1φ1)(φ
2φ2) , (19)
where F 1µν and F
µν are the U(1) field strengths for the ordinary and mirror sectors and φ1
and φ2 are ordinary and mirror electroweak Higgs doublets, respectively. Thus we have only
two extra parameters, ǫ and λ, in addition to those of the ordinary SM. The U(1) kinetic
mixing term will not be of interest to us here.
The most general tree-level Higgs potential of the scale invariant mirror model can be
expressed as
V0(φi) = λ
1φ1 + φ
1φ1 + φ
. (20)
This potential is bounded from below if λ + δ ≥ 0 and λ + δ
≥ 0. We take the Higgs field
φi of each sector in unitary gauge and express them as
, φ2 =
. (21)
4There is a related model where the interchange symmetry is just an internal Z2, unrelated to parity. In
that version, the “shadow” weak interactions are left-handed.
Then the potential (20) takes the form,
V0(r, ω) =
λ+ δ(cos4 ω + sin4 ω)
. (22)
As in the previous section, we can minimise the potential, assuming r 6= 0, which leads to
two cases (depending on the sign of λ):
Broken − P case for λ > 0 : sinω = 0 or cosω = 0 requiring λ+ δ = 0;
Unbroken − P case for λ < 0 : sinω = cosω = 1√
requiring λ+
= 0. (23)
As before, the relations between λ and δ will be satisfied by an appropriate choice of the
renormalisation point µ = Λ. For the broken-P case, we have either 〈φ1〉 or 〈φ2〉 being zero.
Since we are identifying sector 1 as the ordinary sector, we shall concentrate on the sinω = 0
configuration.
Next we calculate tree-level mass-squared matrix at µ = Λ:
Broken− P Case : M2ij = λ〈r〉2
Unbroken − P case : M2ij = λ〈r〉2
The above mass matrices each have vanishing determinant, so one of the physical scalar
fields, the PGB, is massless at tree level. As usual, the conformal anomaly means it obtains
a relatively small mass from the 1-loop correction to V0.
Also, it can be easily seen that the matrices in Eq.(24) have non-negative eigenvalues
provided that λ > 0 (λ < 0) for the broken- (unbroken-) P cases. Thus, there are only two
physically distinct minima of the potential (22):
〈r〉 ≡ v ≈ 246 GeV, 〈cosω〉 = 1, if λ > 0 , (25)
〈r〉 ≡
2 · 246 GeV, 〈cosω〉 =
, if λ < 0 . (26)
In case (25) the electroweak symmetry is broken in the SM sector, while the electroweak
symmetry in the mirror sectors is intact5, and hence the Z2 parity symmetry is spontaneously
broken. In this model, the scalar sector consists of the standard Higgs boson, which is
massless at tree level, and a complex mirror-doublet of bosons with mass squared: m2H =
λv2 ≈ λ(246)2 GeV2.
5SU(2) ⊗ U(1) electroweak symmetry in the mirror sector is eventually broken through mirror SU(3)
quark condensation. For details, see Ref. [9].
In case (26), the SU(2)×U(1) symmetry is broken in each sector so the Z2 mirror sym-
metry remains exact. Consequently, the gauge bosons and fermions in each sector obtain
the same masses as the corresponding particles in the SM. At tree level there is one mas-
sive scalar, with m2H = −4λv2 ≈ −4λ(246) GeV2, and one massless state. This massless
state corresponds to the PGB of broken scale invariance. The mass eigenstates are parity
eigenstates, maximal superpositions of the ordinary and mirror physical Higgs bosons.
Let us now calculate the mass of the PGB boson for each pattern of symmetry breaking.
As briefly reviewed in section II, the mass of the PGB is, in general, given by Eq.(14).
Applying this to the spontaneously broken mirror symmetry case (25) we obtain,
m2h ≃
8π2v2
6m4W + 3m
Z + 4m
H − 12m4t
2π2v2
for mh
>∼ 115 GeV, (27)
while for the unbroken mirror symmetry case (26) we find:
m2h ≃
16π2v2
12m4W + 6m
H − 24m4t
16π2v2
for mh
>∼ 115 GeV. (28)
Thus, we effectively have a mass relation between the light and heavy Higgs bosons in the
models. Numerically, we obtain the approximate relations:
115 GeV
360 GeV Broken case (Eq.25);
115 GeV
600 GeV Unbroken case (Eq.26). (29)
Phenomenologically, the broken mirror symmetry case, Eq.(25), mimics the SM. The
light Higgs, h, couples in exactly the same way as does the SM Higgs field, while the heavier
states, H , couple to the mirror sector. In this case, the experimental lower limit on mh is
the same as the SM limit of approximately 115 GeV. Also, the upper bound on mh inferred
from precision electroweak measurements is mh < MEW ≈ 186 [7]. There is no difficulty in
satisfying these constraints.
In the case of the unbroken mirror symmetry, Eq.(26), the light Higgs field h, couples
to both of the sectors, with coupling strength 1/
2 compared with the SM Higgs. Thus
the limit from precision electroweak measurements in this model is given by Eq.(16) with
c2ω = 1/2. However, for MEW ≈ 186 GeV, this bound is inconsistent with the relation,
Eq.(28). We conclude that the Coleman-Weinberg mechanism is consistent with existing
phenomenological bounds only for the broken-P situation6. Below, we therefore discuss the
broken-P case only.
The experimental lower bound on the mass of the SM Higgs boson, mh > 115 GeV, can
be translated into a lower bound on the coupling λ2(Λ) ≡ 2λ(Λ):
λ2(Λ) > 4.2
115 GeV
, (30)
for the broken P case of Eq.(25). To find the domain of validity of perturbation theory we
look, as before, at renormalisation group equations:
= 3λ21 + λ
2 , (31)
= 3λ1λ2 +
λ22 , (32)
where 1
λ1 = λ + δ. As λ2(Λ) is significantly larger than any other coupling in the SM we
can approximately neglect top-quark and gauge boson contributions. These equations must
be supplemented by the constraint equation (23), which reads
λ1(Λ) = 0 , (33)
for the case (25). Numerically solving these equations, we find that the position, µL, of the
Landau pole is µL ≈ 3 × 102Λ ≈ 105 GeV (for mh at the experimental limit of 115 GeV).
The perturbative domain thus extends confortably above the electroweak scale.
4 Conclusion
We have examined the idea that the standard model Higgs boson might be the pseudo-
Goldstone boson of broken scale invariance. The simplest version of this idea is the original
Coleman-Weinberg model. The problem there is that the mass of the Higgs is too small,
and the spontaneous symmetry breaking can only occur if the top quark is also very light.
However, if there is an extended Higgs sector, then the additional scalar degrees of freedom
can compensate for the heavy top quark, and give a Higgs mass in excess of the experimental
lower limit, currently around 115 GeV.
Specifically, we have considered two phenomenologically consistent models. The first
involves the addition of one (or more) real scalar fields. The additional bosonic degrees
of freedom can lead to phenomenologically successful electroweak symmetry breaking. An
6Of course this conclusion is only for the minimal mirror model with two sectors. In this context it will
be interesting to study the case of generalized mirror models with N-sectors [10]. However, we will leave this
study for the future.
even more constrained symmetry breaking sector arises in the mirror model, which has a
discrete symmetry interchanging the standard model Higgs with a mirror partner. The
discrete symmetry eliminates one parameter in the Higgs potential, and leads to a consistent
electroweak symmetry breaking with the same number of parameters as in the standard
model case.
The proposed scale-invariant models (and their generalizations) can be testable in up-
coming LHC experiments in the case of non-zero cosω. The light PGB Higgs interacts with
Standard Model particles with couplings reduced by factor cosω relative to the correspond-
ing couplings of the Standard Model Higgs boson. In addition heavier Higgs-like boson with
a mass correlated with the mass of PGB Higgs (see Eq.(15) can be observed. The couplings
of this heavy boson with Standard model particles is suppressed by sinω and its total decay
width will be dominated by the decay width into two PGB Higgses. In the case of cosω = 0,
the difference between the PGB Higgs and the Standard Model Higgs is rooted in the scalar
potential, and, most probably, LHC will not be capable to distinguish among them. Future
linear collider (e.g. the proposed ILC) should be able to study this case.
5 Acknowledgements
This work was supported by the Australian Research Council.
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