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Orbifolded
:
.
..: 1110 2007 00093
:
. .
,
2011
7/31/2019 Non-abelian gauge field theories on orbifolded extra dimensions, at zero and finite temperature
2/112
4+1 -
, orbifold S1/Z2,
R
T eV1
Randall-Sundrum. -
,
, -
Wilson , ,
( Hosotani).
-
. ,
, -
,
.
7/31/2019 Non-abelian gauge field theories on orbifolded extra dimensions, at zero and finite temperature
3/112
1 3
2 7
2.1 . . . . . . . . . . . . . . . . . 7
2.2 Higgs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.2 . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Goldstone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Orbifolding 28
3.1 Orbifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 . . . . . . . . . . . . . . . . . . . . . . 30
3.2.1 Orbifold . . . . 30
3.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 R-S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3.1 R-S Orbifold . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 47
4.1 . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 . . . . . . . . . . . . . . . . . . . . . . 504.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.2 . . . . . . . . . . . . . . . 62
4.3 R-S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3.2 . . . . . . . . . . . . . . . 80
5 87
5.1 . . . . . . . . . . . . 875.2 . . . . . . . . . . . . . . . . . . . . . . 91
5.2.1 . . . . . 91
1
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5.2.2 . . . . . . . . . . . . . . . . . 96
6 100
102
SU(2) 104
Randall-Sundrum 106
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1
-
90 , Kaluza Klein (1921), ,
-
, 5- ,
(S1), .
, , 5
, 4- ,
, U(1),
. , ,
, -,
.
3
, -
, , , -
, . , ,
,
,
.
, -
, Randall Sundrum, 10
, ,
, ,
(brane) 4 . ,
3
7/31/2019 Non-abelian gauge field theories on orbifolded extra dimensions, at zero and finite temperature
6/112
(universal extra dimensions), 5 , 5
Randall-
Sundrum. , ,
,
, , ,
Higgs, , , -
, ,
(unitarity).
,
orbifold S1/Z2.
, , ,
, Higgs.
, Randall-Sundrum
(R-S) ,
. ,
,
-
.
: 2 -
, . ,
-
,
Higgs , - U(1) ,
SU(2) U(1), -.
3 -
, ,
2R ( R ) 5 .
orbifold,
orbifold
4
7/31/2019 Non-abelian gauge field theories on orbifolded extra dimensions, at zero and finite temperature
7/112
.
R-S -
.
4 -
, -
. , , 3,
, -Goldstone ,
Higgs . , , , , vev
(vev: vacuum expectation value)
.
R-S
. -
Higgs ( -Goldstone )
,
-
.
5, -
,
. -
.
, 6
-
. , ,
.,
. , -
,
, ,
. , , -
,
5
7/31/2019 Non-abelian gauge field theories on orbifolded extra dimensions, at zero and finite temperature
8/112
.
6
7/31/2019 Non-abelian gauge field theories on orbifolded extra dimensions, at zero and finite temperature
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2
-
,
,
100GeV, . ,
,
,
, , , , -
. , Higgs
(vev),
. , -
, , ,
Goldstone,
vev
Higgs.
2.1 .
G
M. , g
:
g = exp (iiTi) (2.1)
7
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: Ti: , -
, ,
.
, ,
- (2.1)
, :
g = exp (iiTi) = limn
I i i
nTi
n(2.2)
, , , , ,
. , ,
M, . ,
. , , ,
.
, (unitary group) U(1).
U(n) nn A A = 1. , U(1) :
U(1)global = {ei, R} (2.3)
, , U(1) , :
U(1)local = {ei(x), (x) : M R} (2.4)
, (
Minkowski ),
8
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( ), :
(x) (x) = eij(x)tj(x) (2.5)
tj .
,
,
,
.
,
: .
, , ,
U(1),
.
, ,
, :
S= d4x 12() m2 (2.6) ,
U(1). -
U(1). , ,
, ,
, , -
, . , :
ei
= i()ei + ei (2.7)
, , , D,
:
D = ei(x)D (2.8)
, , :
S=
d4x1
2
(D)
D m2
(2.9)
9
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12/112
:
D = igA(x) (2.10)
g,
, A, :
A A = A 1g
(x) (2.11)
, , , -
U(1),
, , -. , ,
.
A , ,
Lorentz. ,
U(1)local.
:
F = A A (2.12)
. ,
:
Sgauge = const. d4xFF
(2.13)
,
.
( -
, Euler-Lagrange, Maxwell)
U(1)local, :
S= d4x 14 FF + 12(D)D m2 (2.14)
10
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,
,
.
, , (
) Christofell
( affine ) .
2
2 : -
. D
A .
-
. , ,
( ).
Minkowski
, , -
,
( ei(x), R). -
:
= limxxo(x) (x
o )
x xo
(2.15)
( , , ,
A1,2,...,n). , ,
, . ( Christofell
)
.
-
, ,
11
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. , Riemann -
:
[Du, Dv]w = R(u, v)w (2.16)
u, v, w (
, [D, D]w = Rw ),
:
R = dA + A A (2.17)
A ,
,
Riemann Einstein-Hilbert, . ,
U(1), :
[D, D] = igF = ig(A A) (2.18)
-
.
SU(n), ,
n n 1, , , n- (
,
, n n n 1), :
S=
d4x
1
4T r[FF
] +1
2
(D)
D m2
(2.19)
: F = (AA)T+gfAAT D = igAT T .
:
= U = eiT (2.20)A A = U AU +
1
ig(U)U
(2.21)
12
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-
SU(n).
, -
,
-
.
, ,
, , - , , , . -
, , ,
Lagrange. ,
, , .
, , -
. , , :
P=
DAeiS[A] (2.22) DA = DAoDA1DA2DA3 .
U(1) -
. :
S[A] = 14
d4xFF
= 14
d4x(A A)(A A)
= 14
d4x
(2A
A 2AA)
= 12
d4x
(A
A) A2A + AA
=1
2
d4xA(
2 )A (2.23) Fourier -
A(x) =
d4k
(2)4A(k)e
ikx :
S= 12
d4k
(2)4A(k)
( k2 + kk)A(k) (2.24)13
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, , ,
A(k) = a(k)k ( A(x) = (x), -
)
P= DA1 = . , ,
. , - ,
( k2 + kk), , , , singular
( )
.
gauge (),
, -
(
, gauge - - -
), ,
. -
(gauge fixing), ,
,
. gauge fixing -
. Faddeev-Popov.
Faddeev-Popov
Faddeev Popov
. , ,
, .
, gauge fixing :
F(A) = 0 (2.25)
, :
G(Aa) = F(Aa) (x) (2.26)
(x) Aa -
14
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:
(A)a = A +
1
ga
+ Aa (2.27)
= A +1
gDa
(2.28)
D
adjoint ( -
).
, , :
1 =
DG (G(Aa)) =
Da (G(Aa))det
G
a
(2.29)
, (
), :
P=
DADa eiS[A](G(Aa))det
G
a
(2.30)
. -
:
,
, -
, , .
, (G(Aa)), :D ei
2
(G(Aa)) (2.31)
:
P=
DADaD ei2
eiS[A](G(Aa))det
G
a
(2.32)
, , gauge - ( P= DAeiS[A], ), :
A = A +
1
gDa
(2.33)
path integral :
P= D
a DAei(S[A]
1(F(A))
2
)detG
a (2.34) , gauge
Lagrange , , ,
, .
15
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R-gauge. -
.
, 2 .
a, ,
a, -
. G a. ,
, :
det
G
a
=
DcDc eic Ga c (2.35)
c c Grassman,
, c1c2 = c2c1.
, , gauge
:
P= const.
DADcDc ei
S[A] 1 (F(A))2+c
G
a
c
(2.36)
, , Faddeev-Popov , -,
. , ( -
) , ,
Lorentz spin -
, .
Faddeev-Popov ghosts. -
, , .
,
ghosts
gauge . , -
ghosts
. U(1)
gauge , Lorentz, :
G(Aa) = A +
1
g2a (x) (2.37)
Ga
=1
g2 (2.38)
16
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ghost :
Sghost =
d4x c(1g
2)c (2.39)
, ,
.
2.2 Higgs
2.2.1
Higgs -
.
, -
, ,
. Higgs -
, U(1)
gauge , .
Higgs
, , vev (-
), ,
, .
.
, Goldstone .
, , .
Higgs U(1) .
, ,
, U(). Lagrangian :
S=
d4x
1
4FF
+1
2(D)
(D) U()
(2.40)
, ,
vev. ,
,
17
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( ,
,
- ). ,
u = 0.
, :
U() = (||2 u2)2 (2.41)
= ueia, a -
. , , (
S1 ).
, , -
. , ,
. ,
2.3
Goldstone. a = 0
, , vev .
,
, Fourier
, :
=
d3p
(2)3/212Ep
(ape
ipx + apeipx
)(2.42)
ap ap , ,
E =
p2 + m2. |0( -
) ap
- . ,
, , v.e.v = 0||0 = 0. , , vev /
Fourier : = 1 + i2 = u. ,
. , ,
, ,
.
:
18
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:
D = 1 + i2 + ieA1 eA2 + ieAu (2.43) (D)(D) = (1)(1) + (2 + euA)2+
+
2eA(12 21) + 2e2A2u1 + e2A2(12 + 22)
(2.44)
:
U(1, 2) = 4u221 (
(12 + 2
2)2
+ 4u1(21 +
22))
(2.45)
,
2 = 0, :
S=
d4x
1
4FF
+ (eu)2AA +1
2
D1D
1 4u221+
+
(2.46)
(, -
ghost, ,
Higgs -
).
: ,
( 1),
mA = eu, U(1)
. , ,
gauge . ,
1 m1 = 2
u.
Higgs .
:
, Goldstone, , Goldstone. ,
, 2. , ,
, gauge
.
, , , -
, 2 spin 1.
Goldstone
.
19
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2.2.2
Higgs U(1)
,
: .
Weinberg-Salam-Glashow,
SU(2)U(1). , , A 22 ( ) 4.
, , , ,
U(1) .
, W, Z0 . Higgs.
,
, SU(2),
2 ,
SU(2) :
= exp (
iaata) (2.47)
ta SU(2). -
Pauli,
.
:
S= d4x
1
4FF 1
4GaGa + (D)
(D) 2 u2
2
(2.48) : D = igA igtaWa , A gauge U(1), Wa gauge SU(2) F, G
a
.
: = u2.
,
, ,
:
o =
0
u
(2.49)
20
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, , -
= o , , . ,
, :
= o + = c1 + ic2
u + h + ic3
(2.50)
, ,
. ,
:
= eitaa(x)
u
0
u + h(x)
(2.51)
, 4
a(x) = 0, h(x) = 0 o. , ,
gauge 3
, :
=
0
u + h
(2.52)
-
:
:
(D)(D) = hh igA(h)(u + h)
igWa
0 h
ta
0
u + h
+ igA(u + h)h + g2A2(u + h)2+
+ gg AWa
0 u + h
ta 0
u + h
+ igWa 0 u + h ta 0h
+
+ gg AWa
0 u + h
ta
0
u + h
+ g2WaWb 0 u + h (tatb)
0
u + h
= hh + g2A2(u + h)2+
+ 2gg AW a0 u + hta
0
u + h
+ g2WaWb 0 u + h(tatb)
0
u + h
(2.53)
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:
2gg A
W1
2
0 u + h
0 11 0
0
u + h
+ W2
2
0 u + h
0 ii 0
0
u + h
+ W3
2
0 u + h
1 00 1
0u + h
= gg AW3(u + h)
2 (2.54)
g2WaWb
0 u + h
(tatb)
0
u + h
=
=1
4
g2WaWb 0 u + h (ab I)
0
u + h =
=1
4g2WaW
a(u + h)2 (2.55)
, :
(D)(D) = (h)
2 + (gA 12
gW3)2
(u + h)2 +1
4g2(u + h)2
(W1W
1 + W2W2)
(2.56)
:
Z = g
g2 + g2
A
gg2 + g2
W3 (2.57)
B = g
g2 + g2
A +
gg2 + g2
W3 (2.58)
W = W1 iW2 (2.59)
g = g/2.
:
(B B)2 + (Z Z)2 + |W+ W+ |2 == (A A)2 + (W3 W3)2 + (W1 W1)2 + (W2 W2)2 (2.60)
. ,
SU(2),
gabcWbWc. , ,
, SU(2) gauge ,
22
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, , ,
.
:
(D)(D) = hh + ZZ(g2 + g
2)(u + h)2+
+ g2W+ W(u + h)2 (2.61)
:
V(h) = 42u2h2 + 4u2h3 + 2h4 (2.62)
, , , Higgs :
S=
d4x
14
(ZZ + BB + W
+W)
+ ZZ(g2 + g2)u2 + g2W+ W
u2 +1
2hh 22u2h2+
+
(2.63)
, , gauge (B), 3
(Z0)
2 - ,
U(1) , , -
- (W ) spin 0 (h) Higgs.
, ,
3 4
( Goldstone
), . , , Higgs -
SU(2)U(1) , .
.
, ,
, Goldstone, Higgs, -
gauge ,
.
23
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2.3 Goldstone.
, , Goldstone,
vev. ,
2 , -
. ,
Fabri-Picasso -
Goldstone.
Hilbert
G ( -
)
Ik, k = 1, 2, . . . , n .
:
= eiakIk (2.64)
, Noether, n -
. ,
:
[jk]
= iL
(i)[Ik]
i
jj (2.65)
n
, , :
Qk =
d3x[jk]
o(x ) = i[Ik]ij
d3xi(x )j (x ) (2.66)
,
:
[i, j] = iij (2.67)
, ,
:
[Qk, m] = [Ik]mjj (2.68)
eiakQkmeiakQk =
eiakIk
mj
j (2.69)
24
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, , , Noether, -
, -
Hilbert . , ,
, ,
.
Fabri-Picasso
Fabri-Picasso : Q Hilbert
, ,
, Q
:
(i) Q|0 = 0
(ii) Q|0 =
Q Hilbert :
Q = d3xjo(x ) (2.70)
, -
, :
[Q, P] = 0 (2.71)
P , .
:
Q|0 = 0|QQ|0 = 0|d3xjo(x )Q|0=
d3x0|eiPxjo(0)eiPxQ|0
=
d3x0|eiPxjo(0)QeiPx|0 (2.72)
, -
:
Q
|0 = d3x0|jo(0)Q|0 (2.73), , :
0|jo(0)Q|0 = const. = c (2.74)
25
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, : c = 0 c = 0, -
:
Q|0 = 0 Q|0 =
Fabri Picasso.
,
,
( , , ): -
(
) Hilbert
( -
).
Goldstone
, Goldstone.
: ,
,
(Q|0 = ) (0|i|0 = ui = 0) |n En 0 |pn| 0, .
vev, (2.68)
:
0|[Qk
, m(0)]|0 = [Ik
]mjuj = 0 (2.75)
d3x
0|[jk]o(x)m(0)|0 0|m(0)[jk]o(x)|0
= 0
d3xn
0|eiPx [jk]o(0)eiPx|nn|m(0)|0 0|m(0)|nn|eiPx[jk]o(0)eiPx|0
= 0
d3xn
0|[jk]o(0)|nn|m(0)|0eipx +iEnt eipx iEnt0|m(0)|nn|[jk]o(0)|0
= 0
n
(pn)0|[jk]o(0)|nn|m(0)|0eiEnt eiEnt0|m(0)|nn|[jk]o(0)|0 = const.
= 0
(2.76)
pn = 0
26
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( ) , n
En = 0
.
:
, Goldstone ( ).
Goldstone -
2
, -
Higgs.
5
.
27
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30/112
3
Orbifolding
. ,
orbifold ,
.
5 orbifolded -
. ,
Randall-Sundrum .
,
.
3.1 Orbifold.
(manifolds)
, , ,
, Riemann
. -
, , , -
, .
28
7/31/2019 Non-abelian gauge field theories on orbifolded extra dimensions, at zero and finite temperature
31/112
R3 .
, , -
.
orbifold, , , .
orbifold -
.
, , ,
.
orbifold n Hausdorff ( 2
2
) {Ui}i, , : (i) -
Vi Rn
G (ii) i Vi/G Ui,
orbifold, .
, , orbifold
manifold .
orbifold: orbifold
M/G, M G (-) , (fixed points) ,
: Gx = x, x M. , , .
orbifold
orbifold, S1/Z2. S1 -
R/N, N {x x + 2Rn,n Z} Z2 - : Z2 =
{1,
1
}. ,
, orbifold
S1 Z2
/
29
7/31/2019 Non-abelian gauge field theories on orbifolded extra dimensions, at zero and finite temperature
32/112
, .
2 fixed points, 0 .
3.1: orbifold S1/Z2. 0
S1
0
, orbifolds -
, Riemannian orbifolds.
, , , , -
orbifold.
orbifolds. -
2 , -
, orbifolds Randall-Sundrum.
3.2
3.2.1 Orbifold
4
,
.
orbifolded
M4 S1/Z2, M4 Minkowski. 5
:
ds2 = gMNdxMdxN = dx
dx R2dy2 (3.1)
30
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5 ,
4 . , R
: xM
xo, x1, x2, x3, y
diag(+1, 1, 1, 1).
-
. -
orbifold breaking. ,
.
SU(2) SU(2)
:
S5D =
d5x
(M)(M) m2
(3.2)
5 2
orbifolding
2 , , . ,
2 5
y = 0.
SU(2) , .
, , 2
.
P1 P2
, 0. :
(y + 2) = P1(y) (3.3)
(y) = P2(y) (3.4)
2 2 2 :
(i) Pi Pi = I, i = 1, 2
(ii) P2 = P12 = P2
,
.
31
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2 :
( + y) = P1P2( y) = P3( y) (3.5)
P3 -
y = . ,
2 . - P1
,
2 orbifold (y = 0 y = )
(
).
, -
P2, P3 2 2 (i) (ii). :
P2 = P3 =
1 0
0 1
(3.6)
, ,
,
5 , Fourier :
(x, y) =
n=n=0 Ann1 (x)cos(ny)n=
n=0 Ann2 (x
)sin(ny)
(3.7)
y [0, ] An: Ao = 12R An=0 =
1R
. -
.
5 . -
, (dimensional reduction),
,
. ,
4 . , , :
,
.
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Fourier . ,
Langrangian y , ,
. , , 5-
Langrangian 2 5 ,
4- Langrangian , Fourier modes
2, , , 4 .
, 4- ,
, :
S4D =
d4x
n=0
|n1 |2 (m2 + (
n
R)2)|n1 |2
+
n=1
|n2 |2 (m2 + (
n
R)2)|n2 |2
(3.8)
:
1) , 4
m2 + (n
R)2.
Kaluza-Klein modes
(-modes). KK-modes
,
.
2) , Fourier -
,
SU(2) . ,
, n = 0
. , , -
( SU(2) ).
, U(1) . ,
, . -
, P2, P3,
, t3 SU(2)
. , t1,2
, t3 (
n1 22 ) e
iaI, I
2 2 , U(1) 2
.
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3.2.2
, ,
orbifold ,
. -
( )
.
SU(2)
, -
, SU(2). , -
orbifold ( y = 0, y = )
. ,
.
SU(2), :
T = (DM)(DM) (3.9)
DM = M
igAaMt
a, (3.4) (3.5)
, :
Aa(x, y)ta = Aa(x, y)P2taP2 (3.10)
Aay(x, y)ta = Aay(x, y)P2taP2 (3.11)
Aa(x, + y)ta = Aa(x
, y)P3taP3 (3.12)Aay(x
, + y)ta = Aay(x, y)P3taP3 (3.13)
, AaM
(x, y)
P2, P3 ta. , ,
, 5
,
2 y
y.
, , ,
.
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A = Aata, :
A(x, y) = PiA(x, y)Pi (3.14)
Ay(x, y) = PiAy(x, y)Pi (3.15)
:
F PiFPi (3.16)Fy PiFyPi (3.17)
:
T r[FMNFMN] = T r[FF
] + 2T r[FyFy]
T r[PiFP
i PiF
Pi ] + 2T r[(
PiFyPi )(
PiF
yPi )]
= T r[FMNFMN] (3.18)
SU(2).
Pauli ta, SU(2)
,
, :
Pit3Pi = t
3
A3yy
y+yA3
A3yyy
y+yA3y
(3.19)
Pit1,2Pi = t1,2
A1,2yy
y+yA1,2
A1,2yyy
y+yA1,2y
(3.20)
, , , - Fourier
. :
A3 =1
2RA3(o) (x
) +n=1
1R
A3(n) (x)cos(ny) (3.21)
A3y =n=1
1R
A3(n)y (x)sin(ny) (3.22)
A1,2 =
n=1
1R
A1,2(n) (x)sin(ny) (3.23)
A1,2y =1
2RA1,2(o)y (x
) +n=1
1R
A1,2(n)y (x)cos(ny) (3.24)
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( ),
5 , 4- :
S4D =
d4x
1
4F3(o) F
3(o) 14 a=1,2,3
n=1Fa(n) F
a(n)+
+ (D(3) A)(D(3)A) +
1
2
a=1,2,3
n=1
(A
a(n)y +
n
RAa(n)
)2+ . . .
(3.25)
: D(3) = i g2RA3(o) A =
A1(o)y iA2(o)y
2,
n = 0 KK-modes ,
, . 4-
.
, ghost
Lagrange.
:
SGF = 12
d5x
a=1,2,3
(Aa yAay)2
= 12
d4x
a=1,2,3
n=0(Aa(n)
n
RAa(n)y )
2(3.26)
5D -
Poincare 5 , -
. , ,
Poincare orbifolding ,
(fixed points).
0, :
S4D = d4x 14 F3(o) F3(o) 14 a=1,2,3
n=1
Fa(n) Fa(n)+
+
a=1,2,3
n=1
(Aa(n) )2 n
R
2+ (D(3) A)
(D(3)A) + . . .
(3.27)
, :
SU(2) 5 , orbifolding -
,
,
U(1) ,
, ( Kaluza-Klein)
mn = n/R.
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,
-
-
orbifold. A,
,
Goldstone , 2 .
-Goldstone .
4- -
Higgs.
SU(2) -
, ,
SU(N) -
.
:
5- SU(N) M4 S1/Z2 P2, P3 . :
1) [ta, Pi] = 0
KK-modes 4-
.
2) [ta, Pi]
= 0
KK-modes 4-
.
3) y-
[ta, Pi ] = 0 , ,
.
, SU(3) (
)
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, :
P2 = P3 =
1 0 0
0 1 0
0 0 1
(3.28)
:
P t1,2,3,8P = t1,2,3,8 (3.29)
P t4,5,6,7P = t4,5,6,7 (3.30)
t1,2,3,8 ,
3 3 , , ,
3 3 , SU(2) U(1) ( U(1) t8). , , orbifold symmetry breaking
:
SU(3)orbifolding SU(2) U(1)
, , 4- ,
.
, SU(3) -
, (H),
modes
, :
2
d4xdy
1
4FayFay =
i g
2R
3a=1
Aata
2 i g
2R
3
2A8
A
2
(3.31)
, , SU(2) U(1), ,
Weinberg, g/g =
3/2.
, , ,
U(1) ,
2, , -
A8M ZM ( ZM U(1)),
2.2.2. ,
SU(3) ( SU(2) toy-model),
, -
.
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, orbifolded ,
.
. , ,
Randall-Sundrum (RS)
RS orbifold .
3.3 R-S.
-
, , orbifold Randall-
Sundrum. , R-S -
U(1) ,
-
. orbifold
R-S : ,
, ,
orbifold -
. ,
R-S anti de Sitter,
AdS/CFT, 4- ( -
) , , ,
, 4-
, Higgs . ,
, R-S
-
.
3.3.1 R-S Orbifold
Randall-Sundrum
Randall-Sundrum, 1999 2 [12]
, ,
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, , 5-,
5 orbifold, , , -
4-
2 orbifold.
.
Einstein, 4- Poincare, - orbifold 2 - 5-
.
. :
ds2 = gMNdxMdxN = e2kR|y|dxdx R2dy2 (3.32)
k =
/24M3
M Planck 5D .
U(1) RS
, ,
, R-S
gauge bulk ( 5 ). ,
, 4-
orbifold
. -
,
5 y . , ,
Fourier, , -
. ,
U(1) R-S
(
)
.
R-S ,
.
: z = 1k
ekRy ,
:
ds2 =1
(kz)2(
dxdx dz2) (3.33)
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, U(1) :
S5D = 14
d4xdz
GGMKGNFMNFN
= 14
d4xdz
1
(kz)5
(kz)4FF 2(kz)4FzFz
=
1
4d4xdz 1
kzFF 2(Az)
2
2(zA)
2 + 4(A)zAzkz
(3.34) -
, 2 , -
( Lagrange).
5 orbifold, -
, 5- Poincare
,
4 . ,
.
, A Az
. , gauge
fixing :
SGF = 1 d
4xdz1
2kz A kzz
Az
kz 2
=
d4xdz1
2kz
(A
)2 + kz
z
Azkz
2 2(A)z
Azkz
(3.35)
, , :
S5D+GF = 14
d4xdz
1
kz(FF +
2
(A
)2 2(Az)2 2(zA)2+
+ 2(1z
Az + zAz)2) (3.36)
Euler-Lagrange , .
, = 1 :
1
kz
2A z( 1
kz(zA)
)= 0 (3.37)
2Az 2zAz + z
Azz
= 0 (3.38)
:
A =n
A(n) (x)(n)(z) (3.39)
Az =n
A(n)z (x)(n)(z) (3.40)
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-
:
2A
(n) + m2nA
(n) = 0
kzz
1kz
z(n)
+ m2n
(n) = 0(3.41)
2A(n)z + m2nA(n)z = 0
2z(n) 1
zz
(n) + 1z2
(n) + m2n(n) = 0
(3.42)
2 -
Minkowski.
, 4-
z,
orbifold. , , (n)(z) (n)(z)
-
(
,
). , A
(n)
,z(x
) KK-modes , ,
spin 1 4- Minkowski.
, ,
5- (
4 )
KK-modes.
, . (n) = zf(n)
:
2zf(n) +
1
zzf
(n) + (m2n 1
z2)f(n) = 0 (3.43)
2z(n) 1
zz
(n) + (m2n +1
z2)(n) = 0 (3.44)
, , Bessel :
(n) = zf(n) = Cnz(J1(mnz) + anY1(mnz)) (3.45)
Cn an -
. ,
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Bessel
Bessel Neumann (J1, Y1). ,
R-S, KK-modes 4-
:
A(x, z) = Cnz n
A(n) (J1(mnz) + anY1(mnz)) (3.46)
Az(x, z) = Cnn
A(n)z
z(n) (3.47)
, , - -
:
dz zJm(kz)Jm(kz) = dz zYm(kz)Ym(kz) =1
k
(k
k) (3.48)
,
,
4- -
orbifold. SU(2)
SU(N).
3.3.2
-
, -
. , . -
:
1) 1: A(x, y) = A(x, y) A(x, y) = A(x, + y) A(x, 0) = 0 A(x, ) = 0 .
2) +1: A(x, y) = A(x, y) A(x, y) = A(x, +y) zA(x, 0) =
0 zA(x, ) = 0 ( yA(x, 0) = 0 yA(x, ) = 0, -).
,
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Bessel:
d
dxYp(kx) = kYp1(kx) p
xYp(kx) (3.49)
an
( ) :
1) 1 y = 0:
an = J1
mnk
Y1
mnk
(3.50)2) +1 y = 0:
an = Jo
mnk
Yomnk
(3.51)
3) 1 y = :
an = J1
mne
kR
k
Y1
mnekR
k
(3.52)4) +1 y = :
an = JomnekR
k
Yo
mnekR
k
(3.53)
2 orbifold. ,
an, , ,
KK-modes.
SU(2)
,
orbifold breaking. SU(2)
, :
P2 = P3 =1 0
0 1 (3.54)
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, :
A3yy
y+yA3
A3yyy
y+yA3y
(3.55)
A1,2
yy
y+y A1,2
A1,2yyy
y+yA1,2y
(3.56)
, o A3 an
:
an = Jo
mnk
Yo
mnk
(3.57)
KK-modes :
Jo
mnk
Yo
mnk
= Jomne
kR
k
Yo
mnekR
k
(3.58) A1,2 :
an =
J1
mnk
Y1mnk
(3.59)J1
mnk
Y1
mnk
= J1mne
kR
k
Y1
mnekR
k
(3.60) 5 ,
(3.46), -
(3.45) ,
.
R-S : KK-modes A3
A1,2. ,
, -
, , , ,
z -
modes, . ,
4- (3.27).
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Kaluza-Klein modes SU(2) -
. , , ,
KK-modes
SU(3). , 1,3,5 2,4,6 .
: m = mka, a = 1015.
3.2: Kaluza-Klein
SU(3) , , Randall-Sundrum.
m = m/ka a = 1015. 1, 3, ..., (2+1)- KK
, 2, 4, ..., 2-
, ,
orbifolding -
(
, KK-modes).
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4
-
-
- orbifold
. -
( KK-modes )
orbifolding
, -Goldstone . , , SU(3), -
4-
. , , ,
, -
Higgs , , -
. ,
, .
, -
, . , ,
, ,
5- , (vev) , -
, Higgs.
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4.1 .
-
,
,
.
, -
.
. - (generating functional) -
(W[J]) :
Z[J] = eiW[J]
=D ei
d4x(
L[]+J(x)(x))
(4.1)
J(x) . , generating functional
J, J = 0 -
(connected correlation functions) . ,
:
0
|
|0
= J=0 =
W
J |J=0 (4.2)
, , J
, -
J J = 0. Legendre
, [J],
, -
(
: generating functional, vev J = 0). , -
:
[J] = W[J[J]]
d4xJ J (4.3)
:
J=
d4y
W[J]
J
J
J
d4yJ
J
J
d4yJ[J](x y)
= J[J] (4.4)
-
, - J = 0,
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vev :
o= 0 (4.5)
Legendre
.
, ,
,
.
. ,
T, ,
(
Legendre ).
, .
,
.
cl = J ( ) (
). path integral
:
Z[J] =
D eiSJ[cl+] (4.6)
SJ
= d4xLJ = d4x(L + J) Legendre,
, , -
.
-
.
Lagrangian :
LJ() = LJ(cl) + L(x) |cl + J(x) + 12 d4y L(x)(y) |cl(x)(y) + . . . (4.7) , ,
, ,
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.
:
Z[J] =
D exp
i
d4x
LJ(cl) + 1
2
d4y
L(x)(y)
|cl(x)(y) + . . .
= (const.) exp id4x(LJ(cl))det
L(x)(y)
|cl
1/2
= (const.) exp
i
d4x
LJ(cl)
+
i
2Trlog
L
(x)(y)|cl
(4.8)
, Legendre, :
[cl] =
d4x
L(cl)
+
i
2Trlog
L
(x)(y)|cl
(4.9)
:
Veff = 1
V T[cl] (4.10)
4.2
4.2.1
3 -
( )
orbifold,
.
4- , -
-Goldstone . ,
5- , Higgs,
, vev -
. , -
, gauge
adjoint .
,
SU(2) SU(3) .
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SU(2),
, A1,2, ,
vev, , A1,2y vev. ,
, U(1) , vev -
A1y. , , 5 :
Ay = Aoy + A
qy (4.11)
: Aoy =
By
0
0
adjoint .
:
DoMAqM = MA
qM ig[AoM, AM] = 0 (4.12)
, , ,
, Lorentz.
, -
, ,
, vev
, , . ,
ghost -
. , ,
, :
14
T r[FMNFMN] = 14
T r[FoMNFoMN] +1
2T r
{AqM
(MN(D
oLD
oL) 2igFoMN)
AqM}
(4.13)
, , ,
vev ( FoMN = 0), gauge
:
Vgaugeeff = i
2d T r
[log
(DoLD
oL)]
(4.14)
d d = 5.
ghost . -
Faddeev-Popov, 2.1, ghost :
Lghost =
d4xc(
DoLDoL)
c (4.15)
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gauge
2. ghost :
Vgauge+ghosteff = i
2(d 2)T r [log (DoLDoL)] (4.16)
( ghost)
T r[
log(
DoLDoL)]
. DoLDoL
:
1) ( adjoint -
) -:
AaM
A1M
A2M
A3M
(4.17)
2) (
AaM)
, , -.
, M = 0, 1, 2, 3 -
,
, DoLDoL :
[DoMD
oM]ab
( + y
y) 0 0
0 ( + y
y g2ByBy) 2gByy0 2gByy (
+ yy g2ByBy)
(4.18)
, , -
. , , -
,
. , -
(3.21) (3.23), 1
Reipx
sin (ny) - A1,2 1
2Reipx
, 1R
eipx
cos (ny) A3.
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:
[DoMD
oM]ab
p2 + ( n
R
)20 0
0p2 + ( n
R
)2+ g2(By)
2
2gBy nR0 2gBy nR
p2 +
(nR
)2
+ g2(By)2 (4.19)
, , :
1 = p2 + n
R
2, 2 = p2 +
n + a
R
2, 3 = p2 +
n a
R
2
: a = gByR. , ,
. ,
, SU(2) 5
, ( - ghosts)
:
Vgauge+ghosteff = 1
2R
3i
2
d4p
(2)4
n=1
log
p2 +
nR
2+
n=
log
p2 +
n + a
R
2+ log(p2)
(4.20)
, , , ,
() vev (a). ,
, vev
(
). :
dD1pE(2)D1
1
2R
n=+
n=log
p2E +
(n x)2R2
= 2(D/2)
(2R)DD2
fD(2x) + x
(4.21)
: fD(x) =
n=1cos(nx)nD
= fD(x + 2) = fD(x), fD(0) = R(D), a ,
:
Vgauge+ghosteff = Cn=1
3
n5cos(2na) (4.22)
: C = 31287R5
SU(3) , -
4
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, , :
1,2 = p2 + n
R
2, 3 = p2 +
n + a
R
2, 4 = p2 +
n a
R
2
5,6 = p2 +
n + a2
R
2, 7,8 = p2 +
n a
2
R
2
, regularization a , :
Vgauge+ghosteff = Cn=1
3
n5
cos(2na) + 2cos(na)
(4.23)
,
adjoint 2
:
(i) P2 = P3 =
1 0
0 1
(ii) P2 = P3 =1 0
0 1
SU(2) , :
(i) P2 = P3 =
1 0 0
0 1 0
0 0 1
(ii) P2 = P3 =
1 0 0
0 1 0
0 0 1
SU(3) . , , ,
-
.
, , 4 ,
4 ,
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Lorentz.
Clifford :
S =i
4[, ] (4.24)
: {, } = I
(Dirac bi-spinors)
Lorentz, ,
,
Clifford . 4 , -
Clifford 4 4 , , , . -
5 , , -
4 4 , 5 = io123 . , -
, M
, i5
Clifford 5 , Dirac , , 4
. 5 Clifford, -
, 5
(
). , ,
, 4-
KK-modes 5- , .
Dirac 4 5-
Clifford
5 (: -
Weyl ),
,
Dirac:
Sfermions =
d5x(
iMDM M)
(4.25)
SU(N), N- Dirac
:
Sfermions =
d5x(
iDM M M)
(4.26)
: DM = MI igAaMta.
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, , -
, , -
y = 0, y = .
, -
. , , Ui1 Ui2
- ( ) i- (i=2,3 0
). :
DM M = I oI igI oAata ++ I oyI 5 igI oAayta 5
yy +yy
Ui1
Ui2o
Ui1 (Ui2) igAaUi1
Ui2o
(PitaPi Ui1) (Ui2)
Ui1 Ui2o yUi1 (5Ui2) + igAayUi1 Ui2o (PitaPi Ui1) (5Ui2) ,
orbifold :
Ui1 = Pi (4.27)
Ui2o5Ui2 = o5 (4.28)
Ui2oUi2 =
o (4.29)
Ui2 5,
Clifford 4 . , 5-
-
orbifold :
Pi 5 (4.30)
, , -
Goldstone . -
,
vev
,
, ,
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1 , :
Zfermions =
DD exp
i
d5x
(iI + gt1 5By
)
= (const.) exp[
log(
det(iI + gt1 5By))]
= (const.) exp1
2T r log (iI
+ gt1 5By)2 (4.31)
, :
Vfermionseff =i
2T r
log
(iI + gt1 5By)2
(4.32)
,
. SU(2) .
,
- :
=
u1R
u1L
u2R
u2L
(4.33)
, :
(iI + gt1 5By)2u1
u2
= 2u1 + igByyu2 + 14g2B2yu12u2 + igByyu1 +
14
g2B2yu2
(4.34), :
(iI + gt1 5By)2 =2 + g2B2y4 igByy
igByy 2 +
g2B2y4
(4.35)
, .
(i) (ii) :
- (i):
- orbifold
,
(R) 1 (L) 2 :
12R
eipx
,1R
eipx
cos(ny) (4.36)
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1
:
1R
eipx
sin(ny) (4.37)
, 4.35 :
Mij p2 + ( nR)2 + a24R2 i anR2 5
i anR2
5 p2 + ( nR
)2+ a
2
4R2
(4.38)
,
:
1,2,3,4 =
n + a
2
R
25,6,7,8 =
n a
2
R
2
, : a = gByR., -Goldstone -
SU(2) :
Vfermionseff = 2iNifund
1
2R
d4p
(2)4
+n=
log
p2 +
n a
2
R
2(4.39)
= 4NifundCn=1
1
n5cos(na) (4.40)
SU(3) , -
. -
.
- (ii)
(R) 1
(L) 2
:
12R
eipx
,1R
eipx
cos
(n +
1
2)y
(4.41)
1 :
1R
eipx
sin
(n +
1
2)y
(4.42)
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, :
Mij p2 +
n+ 1
2
R
2+ a
2
4R2ia(n+ 1
2)
R25
ia(n+ 12 )R2
5 p2 +n+ 1
2
R
2+ a
2
4R2
(4.43)
,
:
1,2,3,4 =
n + a+1
2
R
25,6,7,8 =
n a1
2
R
2, : a = gByR.
,
:
Vfermionseff = 2iNiifund
1
2R d4p
(2)4
+
n=
log p2 +
n a12
R 2
+ log p2 +
n + a+12
R 2
(4.44)
= 4NiifundCn=1
1
n5cos(n(a 1)) (4.45)
SU(3) .
, , , ,
adjoint .
, , SU(2)
SU(3).
Lagrangian:
Lfermions = iM(
M igAbMTb)
M (4.46)
: Tb .
adjoint , , -
. ,
Lagrangian ,
:
, 3 SU(2) ( 8-
SU(3)) , , :
= ata (4.47)
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, adjoint -
, : DM = M igAaM[ta, . . . ], :
Lfermions = iT r[
ctcM(
Mtb igAaM[ta, tb]
)b
] M2aa (4.48) -
, adjoint , :
Vadjfermeff =
i
2T r
log
(iMM + g
5By[t1, . . . ]
)2(4.49)
:(iMM + g
5Aoy[t1, . . . ]
)2= 2 + 2igByy[t
1, . . . ] + g2B2y[
t1[t1, . . . ]]
(4.50)
, ,
:
Mij
2 0 0
0 2 + g2B2y 2gByy0 2gByy 2 + g2B2y
(4.51)
, ,
.
adjoint 2 .
, ,
- orbifold. , ,
,
, , . ,
adjoint
, , , 5
( , ,
) , ,
, Lagrangian
. , i-
(i = 2, 3) :
= ata
Pit
aPi (5a) (4.52)
, , (+) (-)
, 2 -
, .
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- (+):
(+), -
, (R) 3
(L) 1 2
:
12R
eipx
,1R
eipx
cosny (4.53)
3
1 2 :
1R e
ipx
sinny (4.54)
, :
1,2,3,4 = n
R
25,6,7,8 =
n + a
R
29,10,11,12 =
n a
R
2(4.55)
- (-):
(-),
(R) 3 (L) -
1 2
:
1
Reipx
cosn + 12 y (4.56)
3
1 2 :
1R
eipx
sin
n +
1
2
y
(4.57)
, :
1,2,3,4 = n +12
R
2
5,6,7,8 = n + a +12
R
2
9,10,11,12 = n a +12
R
2
(4.58)
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, 2 ,
, :
Vadjfermions(+)eff = 4N
(+)adj C
n=1
1
n5cos(2na) (4.59)
Vadjfermions()eff = 4N
()adj C
n=1
1
n5
cos 2n a +1
2 (4.60)
SU(3)
,
,
2 . SU(3) :
Vadjfermions(+)eff = 4N
(+)adj C
n=1
1
n5(cos(2na) + 2cos(na)) (4.61)
Vadjfermions(
)
eff = 4N()
adj C
n=1
1
n5
cos
2n
a +1
2
+ 2cos (n(a 1)) (4.62) -
, , .
M2, M
.
4.2.2
, -
, ,
orbifolding, SU(2) SU(3) , -
. ,
-Goldstone
, , SU(3) , -
a = gByR, By ,
, -
. :
Veff =3
1287
R5 n=1
1
n5 3cos(2na) 6cos(na) + 4N
(i)fundcos(na) + 4N
(ii)fundcos(n(a
1))+
+ 4N(+)adj cos(2na) + 8N
(+)adj cos(na) + 4N
()adj cos
2n
a 1
2
+ 8N
()adj cos(n(a 1))
(4.63)
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3 (i)
-
a.
4.1: 1 -Goldstone , 3 (i) , -
a = gRB
,
a = 1 By = 1/gR = 0, , , Higgs
2. -Goldstone ,
orbifolding, ,
Higgs vev
.
, , : -
SU(2) U(1) U(1),
U(1) U(1). vev . , -
DoLDoL,
1 , ,
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tree level modes ( U(1))
( 3,5,7), KK-mode
a = 1. n = 1 mode 4
U(1) . ,
-
a = 1. Wilson . -
, SU(3)
:
SU(3)orbifolding SU(2) U(1) radiative
correctionsU(1) U(1) (4.64)
3 adjoint (+)
T .
4.2: 1 -
Goldstone , 3 (+) ,
a = gRB
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, , vev,
1. ,
U(1),
SU(2) adjoint
. :
SU(3)orbifolding SU(2) U(1) radiative
correctionsU(1) (4.65)
3 (ii) 2 adjoint (+)
-
.
4.3: 1 -
Goldstone , 3 (ii) 2
(+) , a = gRB
, U(1)
, Higgs
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. ,
. , ,
:
SU(3)orbifolding SU(2) U(1) radiative
correctionsU(1) (4.66)
Wilson
, -
, .
vev
Wilson ( , , ) ,
.
, , .
Hosotani. , , :
W = T r
Pexp
ig
20
RdyAayta
(4.67)
( A6y
)
vev, A6y = a/gR, path-ordered ,
Wilson, :
Pexp
ig
20
RdyAayta
=
1 0 0
0 cos(a) isin(a)
0 isin(a) cos(a)
(4.68)
a = 0
SU(2) U(1), , - ( ), a = 1
, U(1)U(1) ( ), a = 0, 1 1 ,
U(1).
, -
, orbifold,
,
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, -
2 :
Higgs. , ,
SU(3) , ,
, ,
Higgs Hosotani.
,
. , ,
, , -
,
. :
1) ,
Higgs .
, , Higgs -
fine tuning
weak scale cut-off ,
.
2)
-
:
. ,
Higgs,
.
3)
, .
, , ,
modes
. -
.
, , -
,
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.
,
-
. ,
.
4.3 R-S.
-
adjoint -
, anti de Sitter Randall Sundrum.
Wilson ,
2 .
RS ,
:
, anti de Sitter ,
,
, , -
AdS/CFT.
Higgs. , -
-Goldstone , Higgs
, Higgs.
,
, , , .
,
, -
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, Randall-Sundrum,
-
( .. modes) 2
orbifold (brane localization). -
, Electroweak Precision Tests anti de Sitter RS.
4.3.1
, , -
, -
. , - ,
,
Coleman-Weinberg:
Veff =1
2
r
Nr
d4pE
(2)4log(p2E + m
2r) (4.69)
mr , vev -Goldstone ,
Nr (
- ). , ,
.
-
. , -Goldstone
: A = A + A, , .
, , , -
. ,
vev, vev
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. , A = 0, :
Aa =n
fa(n)(z, h)Aa(n) (x) (4.70)
Aaz = n
zfa(n)(z, h)
mn(h)
Aa(n)z (x) (4.71)
Aa =n
fa(n)(z, h)Aa(n) (x) (4.72)
Aaz = Chhakz +
n
zfa(n)(z, h)
mn(h)Aa(n)z (x
) (4.73)
a a ,
ha
h . Ch
: Ch = g
2kz2z2o . , ,
3, :
fb(n)(z, h)Tb = (z, h)1fb(n)(z, 0)Tb(z, h) (4.74)
: (z, h) = exp(iChhaTak(z2 z2o)/2) = exp ihaTa (z2z2o)fh(z2z2o)
. fh =
1g
2k
z2z2o Higgs.
, , fb(n)(z, 0)
3.3. -
,
( vev
3.3), vev. -
z = zo
,
y = 0 z = zo . - z. , ,
fa(n)(z, 0) : NaC(z, mn) : NaS(z, mn) ( Ni )
SU(3) , (-
) haTa = h6T6,
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:
f4(n)(z, h) = N4S(z, mn)cos(h
2fh) N2C(z, mn)sin( h
2fh) (4.75)
f2(n)(z, h) = N4S(z, mn)sin(h
2fh) + N2C(z, mn)cos(
h
2fh) (4.76)
f5(n)(z, h) = N1C(z, mn)sin(h
2fh) + N5S(z, mn)cos(
h
2fh) (4.77)
f1(n)(z, h) = N1C(z, mn)cos(h
2fh) + N5S(z, mn)sin(
h
2fh) (4.78)
f6(n)(z, h) = N6S(z, mn) (4.79)
f7(n)(z, h) = N7S(z, mn)cos(h
fh) N3
2C(z, mn)sin(
h
fh) +
3
2N8C(z, mn)sin(
h
fh) (4.80)
f3(n)(z, h) =N7
2S(z, mn)sin(
h
fh) +
N3
4C(z, mn)
cos(
h
fh) + 3
+
3
4N8C(z, mn)
1 cos( h
fh)
(4.81)
f8(n)(z, h) = 3
2N7S(z, mn)sin(
h
fh) +
34
N3C(z, mn)
1 cos( h
fh)
+
1
4N8C(z, mn)
3cos(
h
fh) + 1
(4.82)
z ,
,
Ni.
.
. ,
, 4
: (N2, N4), (N1, N5), (N3, N7, N8), (N6). , 3
vev.
:
det(N2, N4) = 0 C(z, mn)S(z, mn)cos2( h2fh
) + C(z, mn)S(z, mn)sin2(
h
2fh) = 0
(4.83)
det(N1, N5) = 0 C(z, mn)S(z, mn)cos2( h2fh
) + C(z, mn)S(z, mn)sin2(
h
2fh) = 0
(4.84)
det(N3, N7, N8) = 0 C(z, mn)
C(z, mn)S(z, mn)cos2(h
fh) + C(z, mn)S
(z, mn)sin2(h
fh)
= 0
(4.85)
, C(z, mn)
vev. , , , () mode, KK
vev . ,
U(1) , vev
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Higgs .
A6 KK-modes vev , ,
mode. vev
, ( 4.4), (-
x = 1015mk
) , -modes
A1,2,4,5 vev:
h2fh .
A3,7 vev
, A8 ,
. KK , , A3
, , A1,2,
mode h2fh
= 0.5 h2fh
= 1
zero modes . , vev
h = fh , U(1)
, mode A8.
, ,
. , , , , :
, ,
KK-modes. ,
() vev 0.01
. , mode (
) : m/m 0, , mode :
V =N
2 d4pE
(2)4log(p2
E
+ (m + m)2) =N
2 d4pE
(2)4log (p2
E
+ m2)1 + 2mmp2E + m2
+(m)2
p2E + m2
=N
2
d4pE
(2)4
log(p2E + m
2) + log
1 +
2mm
p2E + m2
+(m)2
p2E + m2
N2
d4pE
(2)4log(p2E + m
2) (4.86)
, mode , -
vev .
KK-modes .
KK-modes x 20. modes, -
, , (dimensional
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()
()
()
4.4: , 1 , m 1 (), 2 ()
3 () KK mode A1,2,4,5 vev -Goldstone
a = h2fh
R-S73
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regularization). , modified minimal substraction (M S) -
, gauge :
Vgaugeeff =
3
4(4)2
modes
mn(h)4
log
m2nM2
3
2
(4.87)
M -
. ,
-
.
-
Goldstone , ,
, -
. , , Minkowski .
, vielbein, -
Minkowski . vielbein :
g = abeae
b (4.88)
Clifford
Riemannian ,
Minkowski vielbein
( K,M, N Riemann ,
A,B,C Lorentz ). , M = A(e1)MA = AEMA
:
{M, N} = 2gMN I (4.89)
, -
Clifford ,
. , , Dirac Minkowski , -
vielbein .
, , : ea
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. , ea -
Lorentz
. , -
Lorentz.
( ) Minkowski
Lorentz. (spin connection),
, , affine
Christoffel. :
DM = M +1
8AC
CBM[
B, A] (4.90)
(de = 0), - vielbein :
CBM = eCNME
MB + e
CNE
KB
NBK (4.91)
NBK Christoffel.
anti de Sitter ,
vielbein :
eAM =1
kzAM (4.92)
spin connection :
A4M = 4AM = 1
zAM (4.93)
.
, , Randall-Sundrum
:
Sfermion =
d4xdz
1
(kz)5[
iAEMA DM ck]
(4.94)
: m = ck, c .
spin connection vielbein,
-
( 4 ,
5 , , 4 -
. , , ,
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.) : c
z d
dz
fnL = mnf
nR (4.95)
c
z+
d
dz
fnR = mnf
nL (4.96)
: L/R(x) = (kz)2fnL/R(z)nL/R(x
) mn n- KK-mode
.
w = mnz f(n)L/R :
f(n)L/R =
wf
(n)L/R, :
d2f(n)L
dw2+
1
w
df(n)L
dw+ f(
n)L
1
(c 1
2
)2w2
= 0 (4.97)
d2f(n)R
dw2+
1
w
df(n)R
dw+ f
(n)R
1
(c + 1
2
)2w2
= 0 (4.98)
Bessel
:
f(n)R/L(z) = N
J1
2c(mnz) + b
nR/LY1
2c(mnz)
(4.99)
, , -
( ) :
F(n)R/L = N z
52 J12c(mnz) + b
nR/LY1
2c(mnz) (4.100)
bnR/L ,
3, :
bnR/L = J1
2c(mn/k)
Y12c(mnz)
(4.101)
y = 0 :
bnR/L = J 1
2c(mn/k)
Y
12
c(mnz)
(4.102)
y = 0. ,
, , y = 0
CR/L1/2 (mn, z) S
R/L1/2 (mn, z).
( tree level)
y = orbifold. , -
2 , c
( bulk ) 0.5, 0, 25 0, ( 4.5), y = 0, y =
c = 0, 25 ( 4.6):
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()
()
()
4.5: KK modes bulk -
0 , c=0.5 (), c=0.25
() c=0 (). KK modes
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4.6: KK modes bulk -
0 ,
c=0.25. KK modes
Randall-Sundrum, , ,
Higgs . ,
adjoint
() (i/ii) , 4.2.1.
, -
,
KK-modes, vev, -
CR/L1/2 , SR/L1/2
.
adjoint
4.75 4.82 C, S , , C1/2, S1/2. ,
y = ( , -, , Dirac, )
(
),
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:
- (+):
CR
1/2(z, mn)SR1/2(z, mn)cos
2
(h
2fh ) + CR1/2(z, mn)S
R
1/2(z, mn)sin2
(h
2fh ) = 0 (4.103)
: = 0, 1, 1, 2. -
vev .
- (-)
CR
1/2(z, mn)SR
1/2(z, mn)cos2
(
h
2fh ) + CR
1/2(z, mn)SR
1/2(z, mn)sin2
(
h
2fh ) = 0 (4.104)
: = 0, 1, 1, 2.
SU(3),
vev (F(n)(z, h))
vev (F(n)(z, 0)) :
F(n)1 (z, h)
F(n)2 (z, h)
F(n)3 (z, h)
= exp
ih6T6 (z
2 z2o)fh(z2 z2o)
F
(n)1 (z, 0)
F(n)2 (z, 0)
F(n)3 (z, 0)
(4.105)
, y = 0 (
), :
F(n)1L (z, h) = N1LC
L1/2(mn, z) (4.106)
F(n)1R (z, h) = N1RS
L1/2(mn, z) (4.107)
F(n)2L (z, h) = N2LC
L1/2(mn, z)cos(
h
2fh) + iN3LS
L1/2(mn, z)sin(
h
2fh) (4.108)
F(n)2R (z, h) = N2RS
L1/2(mn, z)cos(
h
2fh) + iN3RC
L1/2(mn, z)sin(
h
2fh) (4.109)
F(n)3L (z, h) = N3LS
L1/2(mn, z)cos(
h
2fh) + iN2LC
L1/2(mn, z)sin(
h
2fh) (4.110)
F(n)3R (z, h) = N3RC
L1/2(mn, z)cos(
h
2fh) + iN2RS
L1/2(mn, z)sin(
h
2fh) (4.111)
y =
, (i)
(ii) 4.2.1:
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- (i):
CR1/2(z, mn)SR1/2(z, mn)cos
2(h
2fh) + CR1/2(z, mn)S
R1/2(z, mn)sin
2(h
2fh) = 0 (4.112)
- (ii):
CR1/2(z, mn)SR1/2(z, mn)cos
2(h
2fh) + CR1/2(z, mn)S
R1/2(z, mn)sin
2(h
2fh) = 0 (4.113)
, , -
, Coleman-Weinberg
dimensional regularization M S, -
:
Vfermionseff =
4
4(4)2
modes
mn(h)4
log
m2nM2
3
2
(4.114)
M
.
4.3.2
Higgs , SU(3) ,
, -
. , , 3 -
.
, , bulk 0.25k.
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4.7: 1
-Goldstone vev a = h2fh
, 3
(i) SU(3), R-S
3 (i)
4.7 -
u = h2fh
.
3
u 0.5, , 4.3.1, ,
U(1) (
), U(1) ,
vev.
orbifold -
. , 2 ,
orbifold,
SU(2) , U(1) ., SU(3)
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4.8: 1
-Goldstone vev a = h2fh
, 1
(+) SU(3), R-S
:
SU(3)orbifolding SU(2) U(1) radiative
correctionsU(1) U(1) (4.115)
1 adjoint (+)
T 4.8.
3 adjoint . adjoint
vev fh,
SU(2) U(1) orbifolding U(1) gauge
. , :
SU(3)orbifolding SU(2) U(1) radiative
correctionsU(1) (4.116)
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4.9: 1
-Goldstone vev a = h2fh
, 3
(ii) 2 (+) SU(3), R-S
3 (ii) 2 adjoint (+)
, ( 4.9)
adjoint , U(1)
vev ,
vev Higgs ,
.
. ,
:
SU(3) orbifolding SU(2) U(1) radiativecorrections
U(1) (4.117)
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-
-Goldstone , -
, ,
.
2 , . , Wilson (4.67)
Randal-Sundrum
,
a h/fh, h vev -Goldstone fh -
70 Higgs,
.
Wilson
. , vev 2
, -
,
. ,
bulk RS ,
,
,
orbifolded . ,
, .
RS Higgs
, -
RS , , -
. , ,
AdS/CFT, -
bulk .
bulk -
(primary)
AdS , - bulk .
, , -Goldstone -
- composite, ,
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SU(3) . 5-
, ..
, vev, .
, Higgs
5- -
, .
,
:
Veff =gauge
3
4(4)2
modes
mn(gauge)(h)4
log
m2n(gauge)
M2
3
2
(4.118)
ferm
4
4(4)2 modes
mn(h)4
log m2nM2
3
2
,
:
Veff 30
dy y3log(
gauge(y2)) 4
0
dy y3log(
ferm(y2))
(4.119)
gauge ferm
, . :
(i,)(y2) = C(i)(z, y)S(i)(z, y)cos
2(h
2fh) + C(i)(z, y)S
(i)(z, y)sin
2(h
2fh) (4.120)
C(i), S(i) (i = 1)
(i = 2, 3).
Wronskian Bessel:
J(z)Y(z)
J(z)Y(z) =
2
z
(4.121)
KK ,
:
(i,)(y2) = 1 +
yekR
C(i)(z, y)S(i)(z, y)sin2(
h
2fh) (4.122)
.
, , (distributionfunctions) -
Higgs loop, , -
, Higgs .
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, -
( -
/) Higgs ,
, ,
. ,
RS , , , , -
Higgs, .
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5
SU(3)
, 2 , .
,
SU(3) , , ,
. , -
,
, -
. ,
-
, ,
. ,
. , -
.
5.1 .
,
Z(T), :
Z(T) =r
eEr = T r[
eH]
=
Da|eH|a (5.1)
: = 1/T.
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-
, , , , .
-
,
path integral. ,
, :(xo=0,x )=a(x )(xo=t,x )=b (x )
D eiS = b|eiHt|a (5.2)
(0,x)=(t,x)=a(x )
D eiS = a|eiHt|a (5.3)
, a (
) : = 1T
= it
: ..
D e 1T0 dx
od3xL = T r
[eH
]= Z(T) (5.4)
,
, .
,
( i ) ,
1/T, . -
. ,
, Feynman:
12 =T r
eH12
Z
= F(x1 x2) (5.5)
, , -
. , -
. :
:
(t) = eiHt(0)eiHt (5.6)
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:
()(0) = 1Z
T r[
eH()(0)]
=1
ZT r
[(0)eH()
]=
1
ZT r
[eH(0)( )] = (0)( )
T()(0) = ()(0) = (0)( ) = T( )(0)T()(0) = T( )(0)
, , ,
:
() = ( ) - .. (5.7)
, Fourier :
(x, t ) =n
n(x )eint (5.8)
: n = 2nT n = 2(
n + 12
)T, .
, ,
.
, -
,
, . -
,
. ,
(, ) ,
. -
2, ,
, , .
, :
V = i2
d4p
(2)4log
(p2 + m2) poipo= d4pE(2)4
log(p2E + m
2)
=T
2
n=
d3p
(2)3log
(p2 + (2nT)2 + m2
)(5.9)
, , , , .
. , -
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( )
.
. , ,
( ) ,
n =2n
, :
+n=
f(in) =1
2i
C
dz
2f(z)coth
z
2
(5.10)
C ,
coth(z2
). ()
1 : Re(z) = + 2 : Re(z) = .
:C
dz f(z)coth
z
2
=
1
2
1
dz f(z)ez + 1
ez 1 +2
dz f(z)ez + 11 ez
=1
2
1
dz f(z) +
1
dz f(z)2
ez 1 2
dz f(z) +
2
dz f(z)2
1 ez
=1
2
2
dz (f(z) + f(z)) + 21
dz(f(z) + f(z)) 1ez1
=1
2 C2(f(z) + f(z)) +
C1dz(f(z) + f(z)) 1
ez
1
(5.11)
: C1, C2 1, 2
, .
, ,
. ,
m2. -
. , 2
= p2
+ m2
, :
dV
dm2=
1
2
+n=
d3p
(2)31
2 + 2n
=
d3p
(2)3
1
2
C2
dz
2i
1
2 z2 +C1
dz
2i
1
2 z21
ez 1
=
d3p
(2)31
4+
d3p
(2)31
2
1
e 1
V =
d3p
(2)3
2+
1
ln (1
e) (5.12), :
= i+
dx
2ln(x2 + 2) (5.13)
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-
:
V = Veff(T = 0) + T
d3p
(2)3ln
1 e
p2+m2
(5.14)
-
, .
,
. ( regularization)
:
V
d4p
(2)4log(p2 + m2) (5.15)
,
, Fourier - .
:
V = Veff(T = 0) 4T
d3p
(2)3ln
1 + e
p2+m2
(5.16)
4 .
gauge T = 0
, 3 (, ).
5.2
5.2.1
(
),
. , ,
, , ,
:
V d4p(2)4
log(p2 + m2) (5.17)
, , -
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.
Nd.o.f.T
d3p
(2)3ln
1 e
p2+m2
(5.18)
:
Veff(T) =
n=1
1
n5 3cos(2na) 6cos(na) + 4N(i)fundcos(na) + 4N(ii)fundcos(n(a 1))+
+ N()adj cos(2na) + 8N(+)adj cos(na) + 4N
()adj cos
2n
a 1
2
+ 8N()adj cos(n(a 1))
+
+324
34
+n=
3
0
dp p2 log
1 ep2+(n+a)22
+ 2
0
dp p2 log
1 ep2+(n+a/2)22
4N(i)fund0
dp p2 log
1 + ep2+(na/2)22
4N(ii)fund
0
dp p2 log
1 + ep2+(n(a+1)/2)22
4N()
adj0
dp p2
log
1 + ep2+(n+a+1/2)22 8N()adj
0dp p
2
log
1 + ep2+(n+(a
1)/2)22
4N(+)adj0
dp p2 log
1 + ep2+(n+a)22
8N(+)adj
0
dp p2 log
1 + ep2+(n+a/2)22
(5.19)
-
, -
, .
, , , Kaluza-
Klein modes .
,
, 3
.
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5.1: 1
-Goldstone vev a = gRB, 3 (i) SU(3), RT T=0 (
).
3 (i)
5.1 -
a.
-
U(1) U(1) . ,
SU(2) U(1) 1 ( ,
).
RT 0.155.
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5.2: 1
-Goldstone vev a = gRB, 3 (+) SU(3), RT T=0 (
).
3 adjoint (+)
T 5.2.
( )
SU(2)
U(1) U(1),
Higgs ,
( Weinberg
U(1) - - ).
, 2 -
: 1) , 2 , RT 0.1 a = 1, U(1) U(1) , 2)
1 RT 0.24 orbifolding, SU(2) U(1).
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5.3: 1
-Goldstone vev a = gRB, 3 (ii) 2 (+) SU(3),
RT T=0 ( ).
3 (ii) 2 adjoint (+)
5.3
.
, SU(2)
U(1)
1 ,
RT 0.16. , , , , 1 2 ,
2
. , -
a
-Goldstone
.
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5.2.2
SU(3) -
( , )
5-
.
-
, -
, (
, ),
. -
-
Sakharov. :
1) (
)
2) C CP ( -
)
3) ( -
, CPT -
,
CPT, ).
o
. ,
, U(1)
, , -
( 5)
. , , ,
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, :
JB = nf
g2
322WaWa +
g2
322FF
(5.20)
: Wa SU(2) g
F U(1) -
g. :
F =12
F.
-
SU(2) .
,
, , ( -
: ). :
-
, :
A gg1 (5.21)
g (
S3) ,
. ,
. , gauge
( r = 0
), ,
,
. ,
gg1.
. -
,
(winding number) Pontryagin Chern-Simons
.
:
S3 (
) S3 ( -
Lie - SU(2)) , ,
. f : S1 S1. ,
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, , , ,
, , .
:
1322
d4xT r
WW
(5.22)
, C-S
. -
, , . ,
, . -
WKB,
2 . sphalerons. -
( )
T :
(T) expEsphaleron
T
(5.23)
:
,
T. -
Higgs
vev 0. , -
.
vev 0 vev = 0
( ) - .
, ( -
). ( 1
)
( 2 ),
. sphalerons,
, , -
, . ,
, sphalerons , ,
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,
sphaleron,
wash out. Esphaleron/T 1, Shaposhnikov :
h(Tc)
Tc 1 (5.24)
h(Tc) .
SU(3)
, -
. ,
, CP
,
wash out,
,
.
, 3 -
1 ,
1 2 (
2 ).
,
, .
, vev
3 :
h1(T1)T1
6.45 1g5
h2(T2)T2
4.17 1g5
h3(T3)T3
2.68 1g5
(5.25)
g5 (-
running coupling
constant) 1 -
orbifolded , ,
.
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6
, orbifold,
.
, -
.
orbifold orbifold
. , -
orbifold, ,
. -
(orbifold breaking)
,
.
-
orbifolding. ,
, , -
. ,
, -
-
( Hosotani).
,
adjoint .
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, ,
. ,
vev -
Wilson, 5 ,
,
. RS - Kaluza-Klein modes
5 ,
( ) modes,
orbifold
.
, , , -
.
SU(3) , orbifolding