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General formula of effective potential in 5D SU(N) gauge theory on orbifold
TOSHIFUMI YAMASHITA
Department of Physics, Kyoto University
Kyoto 606-8502, Japan
E-mail: [email protected]
ABSTRACT
We show a simple method to calculate the 1-loop effective potential of the Wilson line in the Hosotani mechanism. This method does not rely on a matrix
representation, but depends on a group theoretical analysis. Thus it is available for more than 5 dimensional theories with any gauge symmetry. As an
application, we show a general formula of the 1-loop effective potentials of 5D
SU(N ) gauge theories compactified on an orbifold, S 1 /Z2 . We also calculate the
effective potentials for some concrete models.
1. Introduction
One of the most interesting scenario in higher dimensional gauge theories is the socalled gauge-Higgs unification scenario. In this scenario, the extra dimensional components of the gauge field, which behave as scalar fields in view of the 4 dimensional theory,
are regarded as the Higgs fields. Thus, we have no need to introduce any fundamental
scalar field. In addition, this scenario has an advantage that the quadratic divergence
does not appear because the higher dimensional gauge invariance protects the mass of the
gauge field. As for the compactification, if the extra dimensions are compactified on a
manifold, the zero modes of the extra dimensional components, which are so-called Wilson
line phases, composes adjoint Higgs. It means that the correct Electro-Weak Symmetry
Breaking can not achieved. Thus, the extra dimensions are usually compactified on an
orbifold, and the orbifold breaking[1] is employed to yield doublet Higgs. Let us illustrate
the orbifold breaking mechanism by a simple example that is a 5 dimensional SU(3) model
compactified on an S 1 /Z2 with radius R. Suppose that the boundary conditions are given
as
Aµ (xµ , −y) = P Aµ (xµ , y)P †,
A5 (xµ , −y) = −P A5 (xµ , y)P †,
Aµ (xµ , πR − y) = P A5 (xµ , πR + y)P † ,
A5 (xµ , πR − y) = −P A5 (xµ , πR + y)P †
(1)
(2)
with P = P = diag(−, +, +) for the 4 dimensional vector Aµ and the scalar A5 , respectively. Then the parities (P, P ) of each components of Aµ and A5 are written as
⎛
⎞
(+, +) (−, −) (−, −)
⎜
⎟
Aµ : ⎝ (−, −) (+, +) (+, +) ⎠ ,
(−, −) (+, +) (+, +)
⎛
⎞
(−, −) (+, +) (+, +)
⎜
⎟
A5 : ⎝ (+, +) (−, −) (−, −) ⎠ .
(+, +) (−, −) (−, −)
(3)
Because only components with parities (P, P ) have zero-modes, we can see that the gauge
symmetry SU(3) is broken down to SU(2)×U(1) and a doublet scalar appears from A5
at the low energy effective theory. If this doublet scalar acquires an appropriate VEV,
the residual SU(2)×U(1) symmetry is further broken down to U(1). Because this mode
is zero-mode, its potential is flat and its VEV is not determined at the tree level. Thus it
is important to calculate the quantum correction for a phenomenological analysis of this
scenario. Conventionally, this calculation has been done by using the structure constant
and thus a hard task for cases of complicated gauge group, such as SO(10) or E6 . Now,
we propose a new method to carry out this calculation more easily.
2. effective potential
2
1-loop effective potential is given by using the inverse of the propagator, ∆−1
f = ∂µ −
D5 (A5 )2 , as
i d4 p 1
−1
∆
(p;
A
)
(4)
Veff (A5 ) = −
tr
ln
5 .
f
2 (2π)4 2πR
To simplify this expression, the eigenvalues of the covariant derivative D5 (A5 )2 must be
aa a
known. For this purpose, let us first define the VEV of A5 as A5 ≡ gR
T , where
g is the gauge coupling constant and T a is the generator. Then we can calculate the
eigenvalues. Conventionally this is done by using the structure constant, but we can find
the eigenvalues without such a troublesome calculation. The point is that D5 (A5 ) for a
field that belongs a representation R is written as
aa
n
(5)
D5 (A5 ) = ∂5 + igR(A5 ) ∼ i + i R(T a ),
R
R
where n is the KK-index and R(T a) is the operation of the generator T a on the representation R. This means that the eigenvalues of the relevant generators give those of D5 (A5 )2 .
The eigenvalues of a generator can be easily found by decomposing R by a subgroup that
contains the generator. Let us illustrate this by the previous SU(3) model. Assuming
only zero-modes can acquire non-vanishing VEVs, we can write the VEV of A5 by one
a
real parameter a as A5 ≡ gR
T 1 , by virtue of the residual SU(2)×U(1) symmetry. This
T 1 is a generator of an SU(2) subgroup of the SU(3). If we decompose a representation of
the SU(3) in terms of representations of the SU(2) subgroup, we can find the eigenvalues
of T 1 . For example, the adjoint representation 8 is decomposed
as 8 → 3 + 2 +
2 + 1,
and we find the eigenvalues Qi (i is the gauge index) as 1, 0, −1; 12 , − 12 ; 12 , − 12 ; 0 . Once
we find the set of eigenvalues Qai of the generators T a , we can write the contribution for
Veff from one bosonic degree of freedom as
Veff (A5 ) = −
i
2
∞
d4 p 1 2
a
2
2
ln
−p
+
(n
−
Q
a
)
/R
.
i a
(2π)4 2πR i n=−∞
(6)
Here, a formula that carries out the integration of the 4-momentum p and the summation
of the KK-index n up to constant term is known and we find
Veff (A5 ) = −
∞
1 C cos(2πnQai aa ),
2 n=1 n5 i
C=
3
.
64π 7 R5
(7)
A significant point is that once we find the set of eigenvalues Qai , we can also find Veff
easily. Because we can find Qai from decomposition of the representation with no need
of any troublesome calculation, this observation makes it possible to calculate Veff easily
even for complicated cases.
3. General formulae of SU(N) theory on S 1 /Z2
By virtue of the new method, we can calculate general formulae of the contribution
for Veff from a field belonging some representation in 5 dimensional SU(N) gauge theories
compactified on S 1 /Z2 with arbitrary N. In Ref.[2], authors showed we can take the
boundary conditions P and P so that they are commute with each other without loss of
generality. Thus it is sufficient to consider the following boundary conditions,
P = diag(+, · · · , +, +, · · · , +, −, · · · , −, −, · · · , −)
P = diag(+, · · · , +, −, · · · , −, +, · · · , +, −, · · · , −),
n+
+
n−
+
n+
−
(8)
(9)
n−
−
−
+
−
where n+
+ + n+ + n− + n− = N. In this case, the parities (P, P ) of Aµ is written as
+⎛
n+
Aµ :
⎜
n+
n−
n+
n−
+
+
−
−
⎞
(+, +) (+, −) (−, +) (−, −)
⎟
⎜
n−
+ ⎜ (+, −)
(+, +) (−, −) (−, +) ⎟
⎟
n+
−
(−, −) (+, +)
n−
−
⎜
⎜
⎜ (−, +)
⎜
⎝
(−, −) (−, +) (+, −)
⎟
⎟.
(+, −) ⎟
⎟
⎠
(10)
(+, +)
Because the parities of A5 are inverse of those of Aµ , the zero-modes of A5 reside in the
(−, −) part in Eq.(10). Thus we parameterize the VEV of A5 as
⎛
⎜
1 ⎜
⎜
A5 =
2gR ⎜
⎝
⎞
0 0 0 Θa
⎟
0 0 Θb 0 ⎟
⎟.
0 Θ†b 0 0 ⎟
⎠
Θ†a 0 0 0
(11)
−
Here Θa is a n+
+ × n− matrix and can be simplified into the following form by the residual
symmetry,
⎛
⎛
Θa =
⎜
⎜
⎜
⎜
⎝
a1 0
0 a2
.. ..
. .
0 0
···
···
..
.
0
0
..
.
· · · aA
0 ···
0 ···
.. . .
.
.
0 ···
0
0
..
.
0
⎞
⎟
⎟
⎟
⎟
⎠
or
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
a1 0
0 a2
.. ..
. .
0 0
0 0
.. ..
. .
0
0
···
···
..
.
···
···
..
.
···
0
0
..
.
⎞
⎟
⎟
⎟
⎟
⎟
⎟
aA ⎟
⎟
0 ⎟
⎟
.. ⎟
⎟
. ⎠
(12)
0
−
+
−
+
−
when n+
+ < n− and n+ > n− , respectively. Here A is the larger of n+ and n− . Θb is similar
−
−
+
as Θa with the replacements (a, n+
+ , n− , A) → (b, n+ , n− , B). For this parameterization,
the contribution from one bosonic degree of freedom of adjoint representational fields is
calculated as
⎡
adj.
Veff
∞
A
B
C 1 ⎣
= −
cos(nπ(a
±
a
))
+
cos(nπ(bi ± bj ))
i
j
2 n=1 n5 i,j
i,j
+2
A B
i
+2 n+
+
+2 n−
+
j
−
−
cos(nπ(ai ± bj − 1))
A
n−
−
i
A
n+
−
cos(nπai ) +
B
cos(nπ(bi − 1))
i
cos(nπ(ai − 1)) +
i
B
cos(nπbi )
.
(13)
i
We can also calculate general formulae for other representations.(See Ref.[3])
4. Phase Structure of 5D SO(10) GUT
As a more phenomenological application, let us consider a 5 dimensional SO(10) GUT,
where boundary conditions break SO(10) into SU(3)×SU(2)×U(1)×U(1). It is known
that zero-modes of A5 appear in this model. And thus many authors consider 6 dimensional theories when they treat SO(10) GUT in higher dimensional theory in order to
project out the zero-modes. But now, we can calculate the quantum correction to the effective potential of the zero-modes and can explicitly show the zero-modes indeed acquire
masses. In addition, we can analyse the effective potential to show the minimum exist at
the point favored phenomenologically.(See Ref.[4])
5. Summary
We showed a simple method to calculate the effective potential of the Wilson line
phase at 1-loop level. The effective potential can be written in terms of the eigenvalues of
generators to which directions A5 acquires non-vanishing VEVs. This method does not
need any troublesome calculation using the structure constant. It makes us possible to
calculate the effective potential much more easily in cases that the gauge group is complicated. By using this method, we calculated general formulae of the effective potential
in 5 dimensional SU(N) gauge theories compactified on S 1 /Z2 . We also showed more
phenomenological application.
6. References
[1] Y. Kawamura, Prog.Theor.Phys. 103 (2000), 613; ibid 105 (2001), 691; ibid 105
(2001), 999.
[2] N. Haba, Y. Hosotani and Y. Kawamura, Prog.Theor.Phys. 111 (2004), 265.
[3] N. Haba and T. Yamashita, JHEP 0402 (2004), 059.
[4] N. Haba and T. Yamashita, Phys.Lett. B597 (2004), 166.