Large Gauge Hierarchy in Gauge-Higgs Unification
TU-788
Large Gauge Hierarchy in Gauge-Higgs Unification
Kazunori Takenaga(a) 1
(a) Department of Physics, Tohoku University, Sendai 980-8578, Japan
Abstract
We study a five dimensional nonsupersymmetric SU(3) gauge theory compacti-
fied on M4 × S1/Z2. The gauge hierarchy is discussed in the scenario of the gauge-
Higgs unification. We present two models in which the large gauge hierarchy is
realized, that is, the weak scale is naturally is obtained from an unique large scale
such as a GUT and the Planck scale. We also study the Higgs mass in each model.
keywords: gauge symmetry breaking, extra dimensions, boundary conditions
1E-mail: takenaga@tuhep.phys.tohoku.ac.jp
Collaboration with M. Sakamoto (Kobe), dragon@kobe-u.ac.jp
Talk given at the SCGT06, International Workshop on ”Origin of Mass and Strong Coupling Gauge
Theories” 21-24 November 2006, Nagoya, Japan
http://arxiv.org/abs/0704.1178v1
1 Introduction
Higher dimensional gauge theory has been paid much attention as a new approach to over-
come the hierarchy problem in the standard model without introducing supersymmetry.
In particular, the gauge-Higgs unification is a very attractive idea [1, 2, 3]. The higher
dimensional gauge symmetry plays a role to suppress the ultraviolet effect on the Higgs
mass. The Higgs self interaction is understood as part of the original five dimensional
gauge coupling, so that the mass and the interaction can be predicted in the gauge-Higgs
unification. The gauge-Higgs unification has been studied extensively [4].
In the gauge-Higgs unification, the Higgs field corresponds to the Wilson line phase,
which is nonlocal quantity. The Higgs potential is generated at the one-loop level after
the compactification. Because of the nonlocality, the Higgs potential never suffers from
the ultraviolet effect [5], which is the genuine local effect, and the Higgs mass calculated
from the potential is finite as well. In other words, the Higgs potential and the mass are
calculable in the gauge-Higgs unification. This is a remarkable feature rarely happens in
the usual quantum field theory. It is understood that the feature entirely comes from
shift symmetry manifest through the Wilson line phase, which is a remnant of the higher
dimensional gauge symmetry appeared in four dimensions. The Higgs mass does not
depend on the cutoff at all, so that two tremendously separated energy scales can be
stable in the gauge-Higgs unification.
We study a five dimensional nonsupersymmetric SU(3) gauge theory, where one of
spatial coordinates compactified on an orbifold S1/Z2. We find two models (model I, II)
which realize the large gauge hierarchy [6].
2 Gauge-Higgs unification
As the simplest example of the gauge-Higgs unification, we study a nonsupersymmetric
SU(3) gauge theory on M4 × S1/Z2, where M4 is the four dimensional Minkowski space-
time and S1/Z2 is an orbifold which has two fixed points, y = 0, πR.
We impose the twisted boundary condition of the field for the S1 direction and at the
fixed points by using the gauge degrees of freedom,
Aµ̂(x, y + 2πR) = UAµ̂(x, y)U
†, (1)
(x, yi − y) = Pi
(x, yi + y), P
i , (i = 0, 1) (2)
where U † = U−1, P
i = Pi = P
i and y0 = 0, y1 = πR and µ̂ stands for µ̂ = (µ, y).
The minus sign for Ay is needed to preserve the invariance of the Lagrangian under these
transformations. A transformation πR + y
P1→ πR − y must be the same as πR + y P0→
−(πR + y) U→ πR − y, so that we obtain U = P1P0. Here we choose P0 = P1 =
diag.(−1,−1, 1).
The gauge symmetry at low energies consists of the zero modes for A(0)a=1,2,3,8µ . We
see that the orbifolding boundary condition Pi breaks the original gauge symmetry SU(3)
down to SU(2) × U(1) at the fixed points [7]. On the other hand, we observe that the
zero mode for A(0)y transforms as an SU(2) doublet, so that we identify the Higgs field as
A(0)4y − iA(0)5y
A(0)6y − iA(0)7y
. (3)
The VEV of the Higgs field is parametrized, by using the SU(2)×U(1) gauge degrees of
freedom, as
〈A(0)y 〉 ≡
= A(0)6y
, (4)
where a is a dimensionless parameter. In order to determine a, one usually valuates the
effective potential for a [2]. The gauge symmetry breaking depends on the values of a0.
It has been known that the matter content is crucial for the correct gauge symmetry
breaking SU(2)× U(1) → U(1)em.
3 Large gauge hierarchy in the gauge-Higgs unifica-
If the Higgs acquires the VEV, the W -boson becomes massive whose mass is given by
MW = a0/2R. This relation defines an important ratio,
= πa0, (5)
whereMc ≡ (2πR)−1. Once the values of a0 is determined as the minimum of the effective
potential, the compactification scale Mc is fixed through Eq.(5). In the usual scenario of
the gauge-Higgs unification, the VEV is of order of O(10−2) for appropriate choice of the
flavor set[8] and this yields Mc ∼ a few TeV. In order to realize the large gauge hierarchy
such as Mc ∼MGUT ,MP lanck, one needs the very small values of a0.
For the very small values of a, the effective potential can be expanded as
V̄eff(a) = −
C(2)(πa)2 +
(πa)4
−ln(πa) +
+ C(4)(ln2)
+ · · · , (6)
where Veff(a) ≡ CV̄eff(a) with C ≡ Γ(52)/π
2 (2πR)5, and the coefficient C(i)(i = 2, 3, 4)
is defined by
C(2) ≡ 24N (+)adj + 4N
adj +
18 + 6dN
adj + 2N
fd + 18N
adj + 3N
, (7)
C(3) ≡ 72N (+)adj + 4N
54 + 18dN
adj + 2N
, (8)
C(4) ≡ 48 + 16dN (+)sadj + 18dN
adj + 2N
adj + 4N
fd + 72N
. (9)
We emphasize that each coefficient in the effective potential is given by the discrete values,
that is, the flavor number of the massless bulk matter. This is the very curious feature of
the Higgs potential, which is hardly seen in the usual quantum field theory, and is a key
point to discuss the large gauge hierarchy in the gauge-Higgs unification.
3.1 Model I
We impose a condition C(2) = 0 in order to obtain the hierarchically small VEV. One
should note that the condition is not the fine tuning of the parameter usually done in
the quantum field theory. This condition is fulfilled by the choice of the flavor set. By
minimizing the Higgs potential, we obtain that
MW =Mc exp
|C(3)|
ln2 +
. (10)
We see that the large gauge hierarchy Mc ∼MGUT ,MP lanck is realized for the large ratio,
C(4)/
∣C(3)
∣ ≫ 1. The magnitude of the ratio for the values of p = 11, 19, where p is
defined by Mc ≡ 10p GeV, is C(4)/
∣C(3)
∣ ≃ 32.54 (p = 11), 59.12 (p = 19). The large
gauge hierarchy is realized if we have C(2) = 0 and the large ratio C(4)/
∣C(3)
∣ at the same
time.
Let us present a few examples of the flavor set in the model I. We choose (k,m) = (1, 0)
as a demonstration. Then, we find that
adj , dN
adj ) = (1, 1), (2, 5), · · · , (N
fd , N
fd ) = (0, 3), (1, 5), · · · .
For (k,m, p) = (1, 0, 19),
adj , dN
adj ) = (0, 29), (1, 33), · · · , (N
fd , N
fd ) = (42, 1), (43, 3), · · · .
For (k,m, p) = (1, 0, 11),
adj , dN
adj ) = (0, 16), (1, 20), · · · , (N
fd , N
fd ) = (22, 1), (23, 3), · · · .
We observe that the flavor numbers dN
adj , N
fd are of order O(10). One has to take care
about the reliability of perturbation theory for such the large number of flavor because
an expansion parameter in the present case may be given by (g24/4π
2)Nflavor, and it must
be (g24/4π
2)Nflavor ≪ 1 for reliable perturbative expansion.
Now, let us study the Higgs mass in the model I. The Higgs mass squared is obtained by
the second derivative of the effective potential evaluated at the minimum of the potential
m2H =
M2W k < M
. (11)
The choice k = 1 is the most desirable one for the large gauge hierarchy, so that the Higgs
mass is lighter than MW , which is the same result in the original Coleman-Weinberg’s
paper [9]. Therefore, one concludes that the large gauge hierarchy and the sufficiently
heavy Higgs mass are not compatible in the model I.
3.2 Model II
We study another model called Model II in this subsection. We introduce massive bulk
fermions [10, 11, 12] in addition to the massless bulk matter in the model I. We introduce
a pair of the fields, ψ+ and ψ− whose parity is different to each other, ψ±(−y) = ±ψ±(y).
Then, a parity even mass term is constructed like Mψ̄+ψ−.
The contribution to the mass term from the massive fermions is given by
ζ(3)C(2)(πa)2 → −1
ζ(3)C(2) + 8NpairB
(πa)2 with
B(2) =
1 + nz +
e−nz,
where Npair stands for the number of the pair (ψ
(+), ψ(−)) and we have defined a dimen-
sionless parameter z ≡ 2πRM = M/Mc. We observe that the potential is suppressed
by the Boltzmann-like factor e−nz, reflecting the fact that the effective potential shares
similarity with that in finite temperature field theory [13].
The essential behavior of the VEV is governed by the factor B(2), i.e. πa0 ≃ γB(2)
with some numerical constant γ of order 1. If we write πa0 = e
−Y , then, one finds,
remembering Eq.(5), that
− Y = ln(πa0)
= (2− p)ln10 ≃
−34.539 for p = 17,
−20.723 for p = 11. (12)
The gauge hierarchy is controlled by the magnitude of Y , in other words, the bulk mass
parameter z, and the large gauge hierarchy is achieved by |z| ≃ 30 ∼ 40. The large gauge
hierarchy is realized by the presence of the massive bulk fermion. We notice that the
flavor number of the massless bulk matter is not essential for the large gauge hierarchy
in the model II.
Now, let us next discuss the Higgs mass in the model II. The Higgs mass is given by
m2H =
−C(3)ln(πa0) +
C(3) + C(4)ln2
M2W F, (13)
where we have defined
F ≡ −C(3)ln(πa0) +
C(3) + C(4)ln2. (14)
The Higgs mass depends on the logarithmic factor. We observe that the larger the gauge
hierarchy is, the heavier the Higgs mass is. An important point is that the coefficient C(3)
is not related with the realization of the large gauge hierarchy, so that it is not constrained
by the requirement of the large gauge hierarchy at all.
In order to demonstrate the size of the Higgs mass in the model II, let us choose
(k, l,m) = (−4,−1,−1). Then, the flavor set is given by
adj , dN
adj ) = (1, 3), (2, 7), · · · , (N
fd , N
fd ) = (3, 1), (4, 3), · · · ,
fd , N
fd ) = (2, 1), (3, 3), · · · , (N
adj , dN
adj ) = (1, 0), (2, 4), · · · .
And the Higgs mass in GeV unit is calculated as
119.5 for p = 17,
92.6 for p = 11,
where we have used g24 ≃ 0.42. Here, we note that in the usual scenario of the gauge-
Higgs unification, one requires g4 ∼ O(1) in order to have the heavy enough Higgs mass
[14, 8]. The large gauge hierarchy enhances the Higgs mass sizably even for the weak
coupling. We observe that for the fixed integers (k, l), the large gauge hierarchy, that is,
large ln(πa0) = −Y enhances the size of the Higgs mass. The larger the gauge hierarchy
is, the heavier the Higgs mass tends to be.
4 Conclusions and discussions
We have studied the five dimensional nonsupersymmetric SU(3) model compactified on
M4 × S1/Z2, which is the simplest model to realize the scenario of the gauge-Higgs uni-
fication. We have discussed whether the large gauge hierarchy is realized in the scenario
or not. The Higgs potential is generated at the one-loop level and is obtained in a fi-
nite form, reflecting the nonlocal nature that the Higgs field is the Wilson line phase in
the gauge-Higgs unification. The Higgs potential is calculable and accordingly, the Higgs
mass, too. We have found two models (model I, II), in which the large gauge hierarchy is
realized. The condition C(2) = 0 is crucial for our discussions.
In connection with the condition, it may be worth mentioning that there are examples,
in which the loop correction is exhausted at the one-loop level (without supersymmetry).
They are the coefficient of the axial anomaly[15] and the Chern-Simons coupling [16].
As for the latter case, a simple reason for the two (higher) loop correction not to be
generated comes from the invariance of the action under the large gauge transformation.
Since the shift symmetry of the Higgs potential can be regarded as the invariance under
the large gauge transformation, one may be able to prove that there is no two (higher)-loop
correction to the mass term of the Higgs potential. In order to confirm it, one needs more
studies of the higher loop corrections to the Higgs potential (mass) in the gauge-Higgs
unification [17].
Acknowledgements
I would like to thank Professor K. Yamawaki and other members of the organizing com-
mittee for their hospitality and for inviting me to participate in SCGT 2006. This work
was supported by the 21st Century COE Program at Tohoku University.
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http://arxiv.org/abs/hep-ph/0504272
http://arxiv.org/abs/hep-ph/0607064
	Introduction
	Gauge-Higgs unification
	Large gauge hierarchy in the gauge-Higgs unification
	Model I
	Model II
	Conclusions and discussions
