non-abelian gauge field theories on orbifolded extra dimensions, at zero and finite temperature

7/31/2019 Non-abelian gauge field theories on orbifolded extra dimensions, at zero and finite temperature

1/112

Orbifolded

:

.

..: 1110 2007 00093

:

. .

,

2011


2/112

4+1 -

, orbifold S1/Z2,

R

T eV1

Randall-Sundrum. -

,

, -

Wilson , ,

( Hosotani).

-

. ,

, -

,

.


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


4/112

5.2.2 . . . . . . . . . . . . . . . . . 96

6 100

102

SU(2) 104

Randall-Sundrum 106


5/112

1

-

90 , Kaluza Klein (1921), ,

-

, 5- ,

(S1), .

, , 5

, 4- ,

, U(1),

. , ,

, -,

.

3

, -

, , , -

, . , ,

,

,

.

, -

, Randall Sundrum, 10

, ,

, ,

(brane) 4 . ,

3


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


7/112

.

R-S -

.

4 -

, -

. , , 3,

, -Goldstone ,

Higgs . , , , , vev

(vev: vacuum expectation value)

.

R-S

. -

Higgs ( -Goldstone )

,

-

.

5, -

,

. -

.

, 6

-

. , ,

.,

. , -

,

, ,

. , , -

,

5


8/112

.

6


9/112

2

-

,

,

100GeV, . ,

,

,

, , , , -

. , Higgs

(vev),

. , -

, , ,

Goldstone,

vev

Higgs.

2.1 .

G

M. , g

:

g = exp (iiTi) (2.1)

7


10/112

: 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


11/112

( ), :

(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


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


13/112

,

,

.

, , (

) Christofell

( affine ) .

2

2 : -

. D

A .

-

. , ,

( ).

Minkowski

, , -

,

( ei(x), R). -

:

= limxxo(x) (x

o )

x xo

(2.15)

( , , ,

A1,2,...,n). , ,

, . ( Christofell

)

.

-

, ,

11


14/112

. , 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


15/112

-

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


16/112

, , ,

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


17/112

:

(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


18/112

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


19/112

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


20/112

( ,

,

- ). ,

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


21/112

:

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


22/112

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


23/112

, , -

= 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)

21


24/112

:

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


25/112

, , ,

.

:

(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


26/112

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


27/112

, , , 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


28/112

, : 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


29/112

( ) , n

En = 0

.

:

, Goldstone ( ).

Goldstone -

2

, -

Higgs.

5

.

27


30/112

3

Orbifolding

. ,

orbifold ,

.

5 orbifolded -

. ,

Randall-Sundrum .

,

.

3.1 Orbifold.

(manifolds)

, , ,

, Riemann

. -

, , , -

, .

28


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


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


33/112

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


34/112

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 . , , :

,

.

32


35/112

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

.

33


36/112

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.

, , ,

.

34


37/112

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)

35


38/112

( ),

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.

36


39/112

,

-

-

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) (

)

37


40/112

, :

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),

, -

.

38


41/112

, 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]

, ,

39


42/112

, , 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)

40


43/112

, 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)

41


44/112

-

:

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 -

. ,

42


45/112

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, -).

,

43


46/112

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)

44


47/112

, :

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).

45


48/112

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).

46


49/112

4

-

-

- orbifold

. -

( KK-modes )

orbifolding

, -Goldstone . , , SU(3), -

4-

. , , ,

, -

Higgs , , -

. ,

, .

, -

, . , ,

, ,

5- , (vev) , -

, Higgs.

47


50/112

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,

48


51/112

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) , ,

, ,

49


52/112

.

:

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) .

50


53/112

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)

51


54/112

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.

52


55/112

:

[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

53


56/112

, , :

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 ,

54


57/112

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.

55


58/112

, , -

, , -

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

,

, ,

56


59/112

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)

57


60/112

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)

58


61/112

, :

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)

59


62/112

, 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 -

, .

60


63/112

- (+):

(+), -

, (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)

61


64/112

, 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)

62


65/112

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 , ,

63


66/112

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

64


67/112

, , vev,

1. ,

U(1),

SU(2) adjoint

. :


correctionsU(1) (4.65)

3 (ii) 2 adjoint (+)

-

.

4.3: 1 -

Goldstone , 3 (ii) 2

(+) , a = gRB

, U(1)

, Higgs

65


68/112

. ,

. , ,

:



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,

,

66


69/112

, -

2 :

Higgs. , ,

SU(3) , ,

, ,

Higgs Hosotani.

,

. , ,

, , -

,

. :

1) ,

Higgs .

, , Higgs -

fine tuning

weak scale cut-off ,

.

2)

-

:

. ,

Higgs,

.

3)

, .

, , ,

modes

. -

.

, , -

,

67


70/112

.

,

-

. ,

.

4.3 R-S.

-

adjoint -

, anti de Sitter Randall Sundrum.

Wilson ,

2 .

RS ,

:

, anti de Sitter ,

,

, , -

AdS/CFT.

Higgs. , -

-Goldstone , Higgs

, Higgs.

,

, , , .

,

, -

68


71/112

, 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

69


72/112

. , 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,

70


73/112

:

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

71


74/112

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

72


75/112

()

()

()

4.4: , 1 , m 1 (), 2 ()

3 () KK mode A1,2,4,5 vev -Goldstone

a = h2fh

R-S73


76/112

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

74


77/112

. , 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 -

. , , ,

75


78/112

.) : 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):

76


79/112

()

()

()

4.5: KK modes bulk -

0 , c=0.5 (), c=0.25

() c=0 (). KK modes

77


80/112

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, )

(

),

78


81/112

:

- (+):

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:

79


82/112

- (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.

80


83/112

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)

81


84/112

4.8: 1


, 1

(+) SU(3), R-S

:


correctionsU(1) U(1) (4.115)

1 adjoint (+)

T 4.8.

3 adjoint . adjoint

vev fh,

SU(2) U(1) orbifolding U(1) gauge

. , :



82


85/112

4.9: 1


, 3

(ii) 2 (+) SU(3), R-S


, ( 4.9)

adjoint , U(1)

vev ,

vev Higgs ,

.

. ,

:

SU(3) orbifolding SU(2) U(1) radiativecorrections

U(1) (4.117)

83


86/112

-

-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, ,

84


87/112

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 .

85


88/112

, -

( -

/) Higgs ,

, ,

. ,

RS , , , , -

Higgs, .

86


89/112

5

SU(3)

, 2 , .

,

SU(3) , , ,

. , -

,

, -

. ,

-

, ,

. ,

. , -

.

5.1 .

,

Z(T), :

Z(T) =r

eEr = T r[

eH]

=

Da|eH|a (5.1)

: = 1/T.

87


90/112

-

, , , , .

-

,

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)

88


91/112

:

()(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)

, , , , .

. , -

89


92/112

( )

.

. , ,

( ) ,

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)

90


93/112

-

:

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)

, , -

91


94/112

.

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

.

92


95/112

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.

93


96/112

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).

94


97/112

5.3: 1

-Goldstone vev a = gRB, 3 (ii) 2 (+) SU(3),

RT T=0 ( ).


5.3

.

, SU(2)

U(1)

1 ,

RT 0.16. , , , , 1 2 ,

2

. , -

a

-Goldstone

.

95


98/112

5.2.2

SU(3) -

( , )

5-

.

-

, -

, (

, ),

. -

-

Sakharov. :

1) (

)

2) C CP ( -

)

3) ( -

, CPT -

,

CPT, ).

o

. ,

, U(1)

, , -

( 5)

. , , ,

96


99/112

, :

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. ,

97


100/112

, , , ,

, , .

:

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 , ,

98


101/112

,

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 , ,

.

99


102/112

6

, orbifold,

.

, -

.

orbifold orbifold

. , -

orbifold, ,

. -

(orbifold breaking)

,

.

-

orbifolding. ,

, , -

. ,

, -

-

( Hosotani).

,

adjoint .

100


103/112

, ,

. ,

vev -

Wilson, 5 ,

,

. RS - Kaluza-Klein modes

5 ,

( ) modes,

orbifold

.

, , , -

.

SU(3) , orbifolding

non-abelian gauge field theories on orbifolded extra dimensions, at zero and finite temperature

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