effective theory of color-octet vector fields
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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
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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
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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 ]
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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
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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
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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
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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
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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
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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)
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L = gsG + gsgAA5
gs g2
gA 1 m2A
M2k
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