Dynamical symmetry breaking in Gauge-Higgs unification on orbifold

0.21 MB·English

by  Naoyuki Haba

hep-ph/0401183
KUNS-1894
February 1, 2008
Dynamical symmetry breaking in Gauge-Higgs unification
on orbifold
Naoyuki Haba1, Yutaka Hosotani2, Yoshiharu Kawamura3,
4
0 and Toshifumi Yamashita4
0
2
1Institute of Theoretical Physics, University of Tokushima, Tokushima 770-8502, Japan
n
2Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
a
J 3Department of Physics, Shinshu University, Matsumoto, Nagano 390-8621, Japan

9 4Department of Physics, Kyoto University, Kyoto, 606-8502, Japan
2

2
v
3
8
Abstract
1
1
We study the dynamical symmetry breaking in the gauge-Higgs unification of the
0
4 5D theory compactified on an orbifold, S1/Z . This theory identifies Wilson line
2
0
degrees of freedoms as “Higgs doublets”. We consider SU(3) SU(3) and SU(6)
/ c × W
h models with the compactification scale of order a few TeV. The gauge symmetries
p
are reduced to SU(3) SU(2) U(1) and SU(3) SU(2) U(1) U(1),
- c L Y c L Y
× × × × ×
p respectively, through the orbifolding boundary conditions. We estimate the one loop
e
h effective potential of “Higgs doublets”, and find that the electro-weak breaking is
: realized through the radiative corrections when there are suitable numbers of bulk
v
i fields possessing the suitable representations. The masses of “Higgs doublets” are
X
(100) GeV in this scenario.
r O
a
1 Introduction
Much attentions have been paid to gauge theories in higher dimensions[1, 2, 3, 4, 5]. (See,
for examples, grand unified theories (GUTs) in higher dimensions on orbifolds.) One of
the strongest motivations of the higher dimensional gauge theory is based on the very
attractive idea that the gauge and the Higgs fields can be unified in higher dimensions[6, 7].
Recently, this possibility has been revisited in Ref.[8, 9, 10, 11, 12, 13, 14, 15, 26]. In these
scenarios the Higgs doublets are identified with the extra-dimensional components of the
gauge fields in higher dimensions. The masses of “Higgs fields” are forbidden by the higher
dimensional gauge invariance. This is the reason why the “Higgs fields” have at most only
finite masses of the order of the compactification scale in these scenarios. The gauge group
in higher dimensions must be larger than the standard model (SM) gauge group in order
to obtain the “Higgs doublets” from the gauge fields in higher dimensions. The gauge
symmetries are reduced by the orbifolding boundary conditions of extra dimensions. The
identification of “Higgs fields” as a part of gauge super-multiplet has been considered in
5D N = 1 supersymmetric (SUSY) gauge theory whose 5th coordinate is compactified on
S1/Z orbifold[9, 10, 11, 12, 13], which corresponds to the 4D N = 2 SUSY gauge theory.
2
Also it is considered in 6D N = 2 SUSY gauge theory, whose 5th and 6th coordinates are
compactified on T2/(Z Z′) orbifold[10], which corresponds to the 4D N = 4 SUSY gauge
2× 2
theory.
In this paper, we consider 5D non-SUSY and N = 1 SUSY theories compactified on an
orbifold, S1/Z , where Wilson line degrees of freedoms are identified as “Higgs doublets”.
2
Quarks and leptons are assumed to be localized on the 4D wall. We consider SU(3)
c
×
SU(3) and SU(6) models with the compactification scale being a few TeV. The gauge
W
symmetries are reduced to SU(3) SU(2) U(1) and SU(3) SU(2) U(1)
c L Y c L Y
× × × × ×
U(1), respectively, through the orbifolding boundary conditions. We estimate the one loop
effective potential of “Higgs doublets”, and find that the electro-weak breaking is realized
through the radiative corrections when there are suitable numbers of bulk fields possessing
the suitable representations. The masses of “Higgs doublets” are (100) GeV in this
O
scenario. The suitable value of sin2θ and the gauge coupling unification are assumed to
W
be realized by the effects of wall-localized kinetic terms, which may not respect the bulk
symmetry∗. We should also assume the baryon number symmetry to avoid rapid proton
decay in the TeV scale compactification.
2 Gauge-Higgs unification on S1/Z
2
We will consider 5D SU(N) gauge theory on S1/Z M4. The gauge fields propagate in
2
⊗
the bulk. The 5th dimensional coordinate (y) is assumed to be compactified on an S1/Z
2
orbifold. Under the parity transformation of Z which transforms y y, the gauge field
2
→ −
A (xµ,y) (M = µ(= 0-3),5) in the 5D space-time transforms as
M
A (xµ, y) = PA (xµ,y)P†, (1)
µ µ
−
A (xµ, y) = PA (xµ,y)P†, (2)
5 5
− −
∗For other possibilities, the power law unification[16] or the acceleratedunification[17] might be useful.
1
where P is the operator of Z transformation. Two walls at y = 0 and πR are fixed points
2
under Z transformation. The physical space can be taken to 0 y πR. Considering
2
≤ ≤
the S1 boundary condition,
A (xµ,y +2πR) = TA (xµ,y)T†,
M M
the reflection around y = πR, Z′, is given by
2
P′ = TP.
The gauge field A (xµ,y) transforms
M
A (xµ,πR y) = P′A (xµ,πR+y)P′†, (3)
µ µ
−
A (xµ,πR y) = P′A (xµ,πR+y)P′†. (4)
5 5
− −
under the parity transformation of Z′. It should be noticed that the signs of parities of A
2 5
are opposite to those of A . According to eigenvalues, ( , ), of parities, (P,P′), the field
µ
± ±
A (xµ,y) is divided into four eigenfunctions as
µ
1 ∞ ny
A (xµ,y) = A(n)(xµ) cos , (5)
µ (+,+) √2δn,0πR n=0 µ (+,+) (cid:18) R (cid:19)
X
1 ∞ (n+1/2)y
A (xµ,y) = A(n)(xµ) cos , (6)
µ (+,−) √πR µ (+,−) R !
n=0
X
1 ∞ (n+1/2)y
A (xµ,y) = A(n)(xµ) sin , (7)
µ (−,+) √πR µ (−,+) R !
n=0
X
1 ∞ (n+1)y
A (xµ,y) = A(n)(xµ) sin . (8)
µ (−,−) √πR µ (−,−) R !
n=0
X
The expansion of A (xµ,y) is done in the same way. The parity eigenvalue of the field
5
A (xµ,y),isoppositeto that ofA . Themassless statessurviving inthelow energy arezero
5 µ
mode components with parity transformation (P,P′) = (+,+). This paper will consider
the situation that zero modes of A are (1,2,1/2) or (1,2, 1/2) under SU(3) SU(2)
5 c L
− × ×
U(1) . We regard these components as “Higgs doublets”, then we call this theory gauge-
Y
Higgs unification. Here the local gauge invariance in the 5D guarantees the masslessness of
the “Higgs field”, so the Higgs mass should be finite after the radiative corrections. We will
studytwomodelsinthefollowingsections. Insection3,weconsiderSU(3) SU(3) gauge
c W
×
theory, where the nontrivial parity operators, P = P′ = diag(1,1, 1) realizes the gauge
−
reduction of SU(3) SU(2) U(1) as well as the “Higgs doublets” appear as the zero
W L Y
→ ×
modesinA [9,10,11]. Insection4,wewillconsider the5DSU(6)theorywiththeZ parity
5 2
operators,P = diag(1,1,1,1, 1, 1)andP′ = diag(1, 1, 1, 1, 1, 1)[10,11]. Inboth
− − − − − − −
models, we estimate the one loop effective potential of the “Higgs doublets” including the
effects of Kaluza-Klein (KK)[18] modes and bulk matter fields. And we study the vacuum
structure of the models and calculate the mass of the “Higgs fields”. We will also study the
SUSY version, where A becomes the imaginary part of an adjoint chiral superfield after
5
dimensional reduction as follows. Since the 5D N = 1 SUSY theory corresponds to 4D
N = 2 SUSY theory, the 5D gauge multiplet,
= (AM,λ,λ′,σ),
V
2
is decomposed to a vector super-field and an adjoint chiral superfield as
V = (Aµ,λ), Σ = (σ +iA5,λ′),
respectively. Then, in the SUSY case, the gauge multiplet transforms as
V(xµ, y) V(xµ,y)
− = P P†, (9)
Σ(xµ, y) Σ(xµ,y)
! !
− −
V(xµ,πR y) V(xµ,πR+y)
− = P′ P′†, (10)
Σ(xµ,πR y) Σ(xµ,πR+y)
! !
− −
corresponding to Eqs.(1)-(4).
3 SU(3) SU(3) model
c W
×
Let us study the possibility of the dynamical symmetry breaking in the SU(3) SU(3)
c W
×
model, where the Higgs doublets can be identified as the zero mode components of A [9,
5
10, 11]. We take
P = P′ = diag(1, 1, 1) (11)
− −
in the base of SU(3) †. Then, they divide A and A as,
W µ 5
(+,+) ( , ) ( , )
− − − −
A = ( , ) (+,+) (+,+) , (12)
µ  
− −
( , ) (+,+) (+,+)
 − − 
 
( , ) (+,+) (+,+)
− −
A = (+,+) ( , ) ( , ) , (13)
5  
− − − −
(+,+) ( , ) ( , )
 − − − − 
 
which suggest SU(3) is broken down to SU(2) U(1) , and there appear one “Higgs
W L Y
×
doublet” in A as the zero mode‡. We assume that the compactification scale, R−1, as a
5
few TeV.
The VEV of A is written as
5
1 λ
a
A = a , (14)
5 a
h i gR 2
a
X
and we can always take VEV as
a = a,
1
and a = 0 for i = 1 by using the residual SU(2) U(1) global symmetry. The effective
i
6 ×
potential of A is given by[9]
5
3 ∞ 1
Vgauge = C [cos(2πna)+2cos(πna)], (15)
eff −2 n5
n=1
X
†As for SU(3) , we take P =P′ =I.
c
‡In SUSY case, there appear two “Higgs doublets” as the zero modes.
3
where C 3/(64π7R5). This means that the point at a = 0 is the minimum in Vgauge,
≡ eff
which suggests SU(2) U(1) is not broken.
L Y
×
Then, in order to realize the electro-weak symmetry breaking, let us introduce extra
fields in the bulk, which are N numbers of complex scalars, φ, and N (N ) numbers of
s f a
Dirac fermions, ψ (ψa), of the fundamental (adjoint) representation. They transform
φ(x, y) = ηPφ(x,y) , φ(x,πR y) = η′P′φ(x,πR+y), (16)
− −
ψ(x, y) = ηPγ5ψ(x,y) , ψ(x,πR y) = η′P′γ5ψ(x,πR+y) , (17)
− −
ψa(x, y) = ηPγ5ψa(x,y)P† , ψa(x,πR y) = η′P′γ5ψa(x,πR+y)P′† , (18)
− −
under parities, respectively. Here η,η′ = , and the effective potential induced from these
±
bulk fields strongly depends on the sign of the product, ηη′. Appendix B shows the bulk
fields’ contributions to the effective potential,
∞ 1 1
Vm = C [2N(+)cos(2πna)+2N(−)cos(2πn(a ))
eff n5 a a − 2
n=1
X
+ (4N(+) N(+) +2N(+))cos(πna)
a − s f
+ (4N(−) N(−) +2N(−))cos(πn(a 1))]. (19)
a − s f −
The index ( ) indicates the sign of ηη′ in Eqs.(16)-(18). Here we denote N = N(+)+N(−),
± s s s
N = N(+)+N(−), and N = N(+)+N(−). Seeing the 1st derivative of V = Vgauge+Vm ,
f f f a a a eff eff eff
each term of ∂V /∂a has a factor sin(πna), which means that the stationary points exist
eff
at least at a = 0 and a = 1§. The difference of the heights between two points is given by
V (a = 0) V (a = 1) =
eff eff
−
∞ 1
2[4(N(+) N(−))+2(N(+) N(−)) (N(+) N(−)) 3]C . (20)
a − a f − f − s − s − (2n 1)5
nX=1 −
This meansthat thesymmetric point ofa = 0 becomes deeper asthenumber ofbulkscalars
with ηη′ = + and bulk fermions with ηη′ = increase. On the other hand, the effects of
−
scalars with ηη′ = and fermions with ηη′ = + make the height of a = 1 decrease. When
−
a = 1 point becomes the vacuum, the Wilson loop becomes
2πR 1 λ
1
W = exp(ig dy a )
C
gR 2
Z0
1
1 λ1 −
= exp(ig 2πR) = 1 , (21)
 
gR 2 −
1
 
 
which suggestsSU(2) U(1) isbroken downtoU(1) U(1). Since theVEVis (R−1),
L Y em
× × O
which is a few TeV, this case can not reproduce the correct weak scale VEV.
Anyhow, in order to realize the suitable electro-weak symmetry breaking, we must find
another vacuum at (0 <)a 1. In this case SU(2) U(1) is broken down to U(1) .
L Y em
≪ ×
Seeing V = Vgauge+Vm in Eqs.(15) and (19), we notice that n = 1 (of the summation of
eff eff eff
§The potential has the symmetry V ( a) = V (a), so that we should only check the region of
eff eff
−
0 a 1.
≤ ≤
4
15
-9.972
10 -9.974
-9.976
5
-9.978
0.2 0.4 0.6 0.8 1 0.02 0.04 0.06 0.08 0.1
-9.982
-5
-9.984
-10
Figure 1: The effective potential in the case of N(+) = 2, N(−) = 8, N(+) = 4, N(−) = 2,
a f s s
and N(−) = N(+) = 0. The unit is C = 3/64π7R5. The horizontal line shows 0 a 1
a f ≤ ≤
and 0 a 0.1.
≤ ≤
n) has dominant contributions for the form of the effective potential. Thus, we can obtain
the suitable value of a (a 1), by introducing bulk fields which induce large coefficients
≪
of cos(πna) and/or cos(πn(a 1)), and small (but non-zero) coefficients of cos(2πna)
− −
and cos(2πn(a 1/2)). We show an example which satisfies the above condition, that
− −
is, N(+) = 2, N(−) = 8, N(+) = 4, N(−) = 2, and N(−) = N(+) = 0. Figure 1 shows the
a f s s a f
V in the region of 0 a 1 and 0 a 0.1. The minimum exists at a = 0.058, which
eff
≤ ≤ ≤ ≤
is around the suitable magnitude of the weak scale in TeV scale compactification. To be
more precise, the kinetic term of the “Higgs field” is obtained from the 5D gauge kinetic
term,
2 dy1Fa Faµ5 = dy1 ∂ Aa +igfaAbAc 2 = ∂ +ig Wατα +i√3g Bµ H 2.
× 4 µ5 2 µ 5 bc µ 5 µ 4 µ 2 4 2
Z Z (cid:16) (cid:17) (cid:12)(cid:12)(cid:18) (cid:19) (cid:12)(cid:12)
Where H = √2πR((A1 + iA2)/√2,(A4 + iA5)/√2)T(cid:12)(cid:12) is the 4D “Higgs doublet”, and(cid:12)(cid:12) g
5 5 5 5 4
is the 4D gauge coupling constant defined as g = g/√2πR. This yields too large sinθ
4 W
and it seems hard to reconcile it with the experimental value through the renormalization
group effect. However, as discussing in Ref.[11], 4D gauge couplings can be also affected
by wall-localized gauge kinetic terms, such as δ(0)λ Fµν2, which do not respect the bulk
0
symmetry. When these couplings dominate bulk gauge couplings¶, we can expect to have
the suitable gauge couplings in the low energy. In this case, the normalization of the low
energy gaugefields might bechangedas(W ,B ) (g /g W ,g /(√3g )B ), which yields
µ µ 2 4 µ Y 4 µ
→
the usual Higgs kinetic term. Thus, we set
a
√2πR A1 = 246GeV.
5 g R ∼
4
D E
The mass squared of the “Higgs field” is given by
∂2V 3g2 ∂2(V /(Cπ2))
m2 = (gR)2 eff = 4 eff . (22)
A5 ∂a2 (cid:12) 32π4R2 ∂a2 (cid:12)
(cid:12)a=0.058 (cid:12)a=0.058
(cid:12) (cid:12)
¶For this situation, we need two ass(cid:12)umptions. One is that the wall-localiz(cid:12)ed Higgs kinetic term is
(cid:12) (cid:12)
negligibly small comparing to the bulk Higgs kinetic term. The other is the bulk induced gauge coupling
shouldbe largerthanwall-localized“gaugecoupling”as,g42 >λ−01. Thus,we shouldtake g4 >1,since the
wall-localized “gauge couplings”, such as λ0, can mainly reproduce the magnitudes of low∼energy gauge
couplings of SU(2)L U(1)Y, (g2,gY), under this situation. And we simply assume g4 = (1) in the
× O
following discussions.
5
By using the approximation formula,
∞ cos(nπξ) (πξ)2 3
ζ(3)+ ln(πξ) (πξ)2, (23)
n3 ≃ 2 − 4
n=1
X
∞ cos(nπ(ξ 1)) 3 (πξ)2
− ζ(3)+ ln2, (24)
n3 ≃ −4 2
n=1
X
for a small ξ, the “Higgs” mass becomes
0.031g 2
m2 4 (130g2 GeV)2, (25)
A5 ∼ R ∼ 4
(cid:18) (cid:19)
where g = (1).
4
O
Now let us consider the SUSY case. Since the effective potential is zero when SUSY
remains, we adopt Scherk-Schwarz (SS) SUSY breaking[19, 20, 21, 22]k, where the mode
expansions are given in Ref.[4]. The effective potential in the SUSY version is given by
∞ 1
Vgauge = 2C (1 cos(2πnβ))[cos(2πna)+2cos(πna)], (26)
eff − n5 −
n=1
X
where β parameterizes SS SUSY breaking. We take β 0.1, since the soft mass is given
∼
by β/R[4]. Since (1 cos(2πnβ)) 1, Eq.(26) means that a = 0 is the minimum point in
− ≥
the effective potential. Therefore, the SU(2) U(1) is not broken as in the non-SUSY
L Y
×
case.
What is going on if the extra fields exist in the bulk? We take the bulk fields of N
f
and N species of hypermultiplets of fundamental (Ψ) and adjoint (Ψa) representations,
a
respectively. The bulk hypermultiplets, Ψ = (φ,φc†,φ˜,φ˜c†) and Ψa = (φa,φac†,φ˜a,φ˜ac†),
which are decomposed into chiral superfields as, Φ = (φ,φ˜), Φa = (φa,φ˜a), and Φc =
(φc,φ˜c), Φac = (φac,φ˜ac), where Φ (Φa) and Φc (Φac) have conjugated transformation
under the gauge group. They transform
Φ(xµ, y) Φ(xµ,y)
− = ηP , (27)
Φc†(xµ, y) Φc†(xµ,y)
! !
− −
Φ(xµ,πR y) Φ(xµ,πR+y)
− = η′P′ , (28)
Φc†(xµ,πR y) Φc†(xµ,πR+y)
! !
− −
Φa(xµ, y) Φa(xµ,y)
− = ηP P†, (29)
Φac†(xµ, y) Φac†(xµ,y)
! !
− −
Φa(xµ,πR y) Φa(xµ,πR+y)
− = η′P′ P′†, (30)
Φac†(xµ,πR y) Φac†(xµ,πR+y)
! !
− −
under the parities, respectively. According to the sign of ηη′ we denote N = N(+) +N(−)
f f f
and N = N(+) +N(−). We always take even number of N(±) to avoid the gauge anomaly.
a a a f
kFor the effective potential in other SUSY breaking, the calculation has benn done in the case of the
S1 compactification in Ref.[23].
6
-2.143
2
-2.144
1
-2.145
0.2 0.4 0.6 0.8 1
-2.146
-1
-2 0.02 0.04 0.06 0.08 0.1
Figure 2: The effective potential in the case of N(+) = N(−) = 2, N(−) = 4, N(+) = 0 with
a a f f
β = 0.1. The unit is C = 3/64π7R5. The horizontal line shows 0 a 1 and 0 a 0.1.
≤ ≤ ≤ ≤
Appendix B suggests the extra matter contributions for the effective potential are given by
∞ 1 1
Vm = 2C (1 cos(2πnβ)) [N(+)cos(2πna)+N(−)cos(2πn(a ))
eff n5 − × a a − 2
n=1
X
(+) (−)
N N
+2(N(+) + f )cos(πna)+2(N(−) + f )cos(πn(a 1))]. (31)
a 2 a 2 −
As the non-SUSY case, the 1st derivative of V = Vgauge +Vm has the factor sin(πna),
eff eff eff
which means the stationary points exist at a = 0 and a = 1. The difference of the heights
between two points is given by
V (a = 0) V (a = 1) =
eff eff
−
(N(+) N(−)) ∞ 1
8[1 (N(+) N(−)) f − f ]C (1 cos(2π(2n 1)β)).(32)
− − a − a − 2 (2n 1)5 − −
nX=1 −
This means that the height of the point at a = 0 becomes high (low) as increasing the
number of N(+) (N(−)) and N(+) (N(−)). Equation (32) is consistent with the results in
a a f f
Ref.[5]. As for the case of N(+) = 1, N(+) = N(−) = N(−) = 0, the effective potential
a f f a
vanishes as V 0, since this case has 5D N = 2 SUSY and there is the residual SUSY
eff
≡
after the SS SUSY breaking[24].
As the non-SUSY case, in order to obtain the suitable value of a (a 1), we should in-
≪
troducebulkhypermultipletswhichinducelargecoefficientsof cos(πna)and/orcos(πn(a
− −
1)) and small (but non-zero) coefficients of cos(2πna) and cos(2πn(a 1/2)). We show an
− −
example which satisfies the above condition, that is, N(+) = N(−) = 2, N(−) = 4, N(+) = 0
a a f f
(Fig.2). In this case with β = 0.1, the minimum exists at a = 0.047, and the “Higgs” mass
squared is given by
0.025g 2
m2 4 (130g2 GeV)2, (33)
A5 ∼ R ∼ 4
(cid:18) (cid:19)
where g = (1). It should be noticed that the numerical analysis shows the value of a
4
O
depends on that of β.
7
4 SU(6) model
We study the vacuum structure of the SU(6) GUT, in which the Higgs doublets can be
identified as the zero mode components of A [10, 11]. We take
5
P = diag(1,1,1,1, 1, 1), (34)
− −
P′ = diag(1, 1, 1, 1, 1, 1), (35)
− − − − −
which divide A and A as
µ 5
(+,+) (+, ) (+, ) (+, ) ( , ) ( , )
− − − − − − −
(+, ) (+,+) (+,+) (+,+) ( ,+) ( ,+)
 − − − 
(+, ) (+,+) (+,+) (+,+) ( ,+) ( ,+)
A =  − − − , (36)
µ  (+, ) (+,+) (+,+) (+,+) ( ,+) ( ,+) 
 − − − 
 ( , ) ( ,+) ( ,+) ( ,+) (+,+) (+,+) 
 
 − − − − − 
 ( , ) ( ,+) ( ,+) ( ,+) (+,+) (+,+) 
 − − − − − 
 
( , ) ( ,+) ( ,+) ( ,+) (+,+) (+,+)
− − − − −
( ,+) ( , ) ( , ) ( , ) (+, ) (+, )
 − − − − − − − − − 
( ,+) ( , ) ( , ) ( , ) (+, ) (+, )
A =  − − − − − − − − − . (37)
5  ( ,+) ( , ) ( , ) ( , ) (+, ) (+, ) 
 − − − − − − − − − 
 (+,+) (+, ) (+, ) (+, ) ( , ) ( , ) 
 
 − − − − − − − 
 (+,+) (+, ) (+, ) (+, ) ( , ) ( , ) 
 − − − − − − − 
 
They suggest that P and P′ make SU(6) broken to SU(3) SU(2) U(1) U(1). Also,
c L Y
× × ×
there appear one “Higgs doublet” in A as the zero mode. As in the previous section, the
5
compactification scale is assumed to be of order a few TeV. The VEV of A is written in
5
the similar way as Eq.(14) as
1 λ
16
A = a (38)
5
h i gR 2
by using the residual SU(2) U(1) global symmetry. The calculation in the appendix A
×
suggests the gauge part of the effective potential in the SU(6) model as
3 ∞ 1
Vgauge = C [6cos(πn(a 1))+2cos(πna)+cos(2πna)]. (39)
eff −2 n5 −
n=1
X
From this equation, we can easily show that
∞ 1
Vgauge(a = 0) Vgauge(a = 1) = 12C > 0. (40)
eff − eff (2n 1)5
nX=1 −
The numerical calculation of the effective potential shows us that a = 1 (a = 0) point is
the global (local) minimum. The vacuum at a = 1 has the Wilson loop
2πR 1 λ
16
W = exp(ig dy a )
C
gR 2
Z0
1
−
1
 
1 λ 1
16
= exp(ig 2πR) =  , (41)
gR 2  1 
 
 1 
 
 − 
 1 
 
 
8
15
-1.0836
10
-1.0838
5
0.02 0.04 0.06 0.08 0.1
0.2 0.4 0.6 0.8 1 -1.0842
-5
-1.0844
-10
Figure 3: The effective potential in the case of N(+) = N(−) = 2, and others being zero.
a f
The unit is C = 3/64π7R5. The horizontal line shows 0 a 1 and 0 a 0.1.
≤ ≤ ≤ ≤
which suggests that SU(2) U(1) is broken down to U(1) U(1). This means that
L Y em
× ×
the vacuum at a = 1 is not suitable, since the weak scale becomes too large. It can be
possible to assume that we exist at a = 0 in the early universe, and the life time of this
false vacuum is longer than the universe history[4]. However, the suitable electro-weak
symmetry breaking can not be realized, anyway.
Then, let us introduce the extra fields in the bulk for the suitable dynamical symmetry
breaking ofSU(2) U(1) . Weintroducethebulkfields ofN numbers ofcomplex scalars,
L Y s
×
φ, and N (N ) numbers of Dirac fermions, ψ (ψa), of the fundamental (adjoint) repre-
f a
sentation. The transformations under parities are the same as Eqs.(16)-(18). Appendix B
suggests that the effective potential induced from the bulk fields is given by
∞ 1 1
Vm = C [2N(+)cos(2πna)+2N(−)cos(2πn(a ))
eff n5 a a − 2
n=1
X
+(4N(+) +12N(−) +2N(+) N(+))cos(πna)
a a f − s
+(12N(+) +4N(−) +2N(−) N(−))cos(πn(a 1))]. (42)
a a f − s −
The index ( ) shows the numbers of bulk fields with the sign of ηη′ in Eqs.(16)-(18). We
±
denote as N = N(+)+N(−), N = N(+)+N(−), and N = N(+)+N(−). As in the previous
s s s f f f a a a
section, the 1st derivative of V = Vgauge+Vm suggests that the stationary points exist
eff eff eff
at a = 0 and a = 1. The difference of the heights between two points is given by
Veff(a = 0) Veff(a = 1) =
−
∞ 1
2[6 8(N(+) N(−))+2(N(+) N(−)) (N(+) N(−))]C . (43)
− a − a f − f − s − s (2n 1)5
nX=1 −
The symmetric point of a = 0 becomes deeper (higher) as N(+), N(−), N(+) (N(−), N(+),
a f s a f
N(−)) increase.
s
As the previous section, in order to obtain the suitable value of a (a 1), we should
≪
introduce bulk fields which induce large coefficients of cos(πna) and/or cos(πn(a 1))
− −
and small (but non-zero) coefficients of cos(2πna) and cos(2πn(a 1/2)). We show an
− −
example which satisfies the above condition, that is, N(+) = N(−) = 2, and others being
a f
zero (Fig.3). In this case, the minimum exists at a = 0.072, and the “Higgs” mass squared
is given by
0.038g 2
m2 4 (130g2 GeV)2, (44)
A5 ∼ R ∼ 4
(cid:18) (cid:19)
9