Effective Theory Of Color-Octet Vector fields

7/31/2019 Effective Theory Of Color-Octet Vector fields

1/13


2/13

L = 14

i

FaiFai +

i j

1

2AaiM2ijAaj

Fai = Aai Aai gifabcAbiAci

a i = 1 . . . N

Aai =

fabcAbi

c

1

gi

a

L =i j

AaiM2ij1

gja


3/13

Lj

M2ij 1gj = 0

M2i M2ii = gij=i

M2ij1

gj

v1 =1i1

g2i

1

g1...1

gN

M2

G1 = g

iAigi

g1

g2i

1

g2i

G1G1

Ga1 = fabcGb1c 1

ga

G1M2

vi i = 2 . . . N )

Gi =j

vijAj

i = 2 . . . N j = 1 . . . N vij

j

vij1

gj= 0


4/13

G1

Gi

Gai = j

vijfabcAbj

c j

vij1

gja

Gai = fabcGbic i = 2 . . . N Gi i = 1

L = (iD m)D

D + i a

2 jgjfjA

aj

i

fi = 1

Aaj fj

Dj = + ia

2gjA

aj

D =

j

fjDj

{fj}Dj

igja

2Faj [Dj, Dj ]


5/13

G1

L = L0 + L1 + L2

L0 = (i m)

L1 = 1 a

2Ga1

L2 = j a

2Gaj j = 2 . . . N

i i = 1 . . . N )

L2 = 0L1

1 = gG1

fi

1 = g Gi

L = (i m) a

2

i

g

gi+j

jvji

Aai

fi 1gi

g

gi+j

jvji

fii fi = 1

N = 2


6/13

L = 14

Fa1Fa1

14

Fa2Fa2 +

M2

2g2

g1A

a1 g2Aa2

2M

G1 = A1 cos() + A2 sin()G2 = A1 sin() + A2 cos()

sin() = g/g2 g

g =g1g2g21 + g

22

.

mG1 = 0

mG2 =M

cos()

G2 G1G1

LG2G1G1 =g1 cos2()sin() + g2 cos()sin2() fabc{Ga1Gb1 Gc2 ++G

a1G

b2 G

c1 + G

a2G

b1 G

c1 }

g1 cos() = g2 sin()G2G1G1

L =

i m gGa1a

2 (g

22f2 g21f1)

g21 + g22

Ga2a

2


7/13

g1 = g2 f1 = f2 G2

G2 G1G1G2G2 G1G1G2G2

G2G2

f1 f2G2

f2 = 0 f1 = 1

N = 2

v2tr

DUDU

v

DU

DU = U

i

2g1A1U

i

2g2A2U

A1 A2U = 1

g2 g1

m21 =2g1v

2

1 + g2v2

2M2

m22 = M2 1 + g2v

2

2M2


8/13

tan() = g1g2

1

g2v2

2M2

1 + g2v2

2M2

G2 G1G1

g1 cos() = g2 sin()tan()

LG2G1G1 = 2g2

1v2

2M2+g2v2

1 g2v2

2M2

1+g2v

2

2M2

2fabc{Ga1Gb1 Gc2 +

+Ga1G

b2 G

c1 + G

a2G

b1 G

c1 }

L = (iD m)

D = + ia

2(1 f)g1A21 + i

a

2fg2A

22

[g1 cos() + f(g2 sin() g1 cos())] Ga1a

2

g2 sin() g1 cos() = 0

f

N

M

2

i = gi

gi1M2

ii1 gi

gi+1M2

ii+1

M210 = M21N

M2NN+1 = M2N1


9/13

M2

ii1 = M2

ii+1 = k

2

k

2

M2i = 2k2

M2 = k2

2 1 0 0 . . . 0 11 2 1 0 . . . 0 00 1 .. 0 .. . .. . .. . .

0 . .1 0 2

N 1

tt

SU(3)LSU(3)Rl L SU(3)L r R

SU(3)R

SU(3)L SU(3)R SU(3)c

L = 12

tr {GLGL } 1

2tr {GRGR }

12

tr {LL } 1

2tr {RR }

+M2

g2tr(gl g

L)2

+M2

g2tr(gr g

R)2

+

f2

2tr

DUDU

G

A


10/13

GL = l

l

igL [l, l]GR = r r igR [r, r]L = L L igL [L, L]R = R R igR [R, R]DU = U igLlU + igRUr

U (3, 3) SU(3)L SU(3)RSU(3)c

g gg/g

mG = 0

mA =gf2

mG = MmA = M

g/g

G =12

l +12

r +g2g

L +g2g

R

A = 12

l +12

r g2g

1 m2A

M2

L +

g2g

1 m2A

M2

R

G =g2g

l +g2g

r 12

L 12

R

A = g2g

1 m2AM2

1l +

g2g

1 m2A

M2

1r +

12

L 12

R

l L SU(3)L r R SU(3)Rg(1k)l+gkL g(1k)r+gkR k

k

L = 12

g(1 k)l(1 5) + 12

gkL(1 5) +

+1

2g(1 k)r(1 + 5) + 1

2gkR(1 + 5)


11/13

L = gsG + gsgAA5

gs g2

gA 1 m2A

M2k


12/13


13/13

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Effective Theory Of Color-Octet Vector fields