T
3
gauge transformation unbroken — massless photon field
W
μ
3
— charged
W
±
μ
= (
W
μ
1
±
iW
μ
2
)
/
√
2
with mass
gf
— neutral scalar “Higgs” particle
φ
3
with mass
√
2
λf
charges of other particles quantized — must appear in
SO
(3)
representations
16
SU
(2)
with a doublet of scalars
L
(
φ, W
) =
D
μ
φ
†
D
μ
φ

V
(
φ
)

1
4
W
μν
a
W
aμν
T
a
=
τ
a
2
[
T
a
, T
b
]
=
i ²
abc
T
c
D
μ
φ
=
∂
μ
φ
+
ig
τ
a
2
φ
W
μν
a
=
∂
μ
W
ν
a

∂
ν
W
μ
a

g²
abc
W
μ
b
W
ν
b
V
(
φ
) =
λ
2
(
φ
†
φ

f
2
/
2)
2
⇒ h
0

φ

0
i
=
0
f/
√
2
¶
conventional choice
possible because all
directions are equivalent
L
(
φ, W
) =
1
2
∂
μ
φ
†

ig
‡
0
f/
√
2
·
W
μ
a
τ
a
2
¶
∂
μ
φ
+
igW
bμ
τ
b
2
0
f/
√
2
¶¶

1
4
W
μν
a
W
aμν
+
cubic and
higher terms
‡
0
f/
√
2
·
W
μ
a
τ
a
2
W
bμ
τ
b
2
0
f/
√
2
¶
=
f
2
8
W
μ
a
W
aμ
suggests
another
symmetry
17
SU
(2)
with a doublet of scalars — in
σ
model language
φ
=
Σ
√
2
0
1
¶
Σ =
σ
+
i~
τ
·
~π
Σ
†
Σ = ΣΣ
†
= (
σ
2
+
~π
2
)
I
h
0

Σ

0
i
=
fI
L
(Σ
, W
) =
1
2
(0 1)
D
μ
Σ
†
D
μ
Σ
0
1
¶

V
(Σ)

1
4
~
W
μν
·
~
W
μν
=
1
4
tr
(
D
μ
Σ
†
D
μ
Σ
)

V
(Σ)

1
4
~
W
μν
·
~
W
μν
gauged
SU
(2)
acts on the left
D
μ
Σ =
∂
μ
Σ +
ig
~
W
μ
·
~
τ
2
Σ
RH
SU
(2)
is a
global symmetry
δ
Σ =
i~
²
L
·
~
τ
2
Σ

i
Σ
~
²
R
·
~
τ
2
~
²
L
=
~
²
R
is unbroken for
h
0

Σ

0
i
=
fI
because
[
~
τ, I
]
= 0
L
(Σ
, W
) =
1
2
‡
∂
μ
σ

i~
τ
·
(
∂
μ
~π
+
gf
~
W
μ
/
2)
·‡
∂
μ
σ
+
i~
τ
·
(
∂
μ
~π
+
gf
~
W
μ
/
2)
·

λf
2
2
σ
2

1
4
~
W
μν
·
~
W
μν
scalar with mass
√
λf
vectors with mass
gf/
2
three gauge and three global generators are broken  but three combinations are
unbroken — so there were three GB to be eaten to become the longitudinal
components of the triplet of massive gauge bosons
18
SU
(2)
×
SU
(2)
coupled to
σ
model
Σ =
σ
+
i~
τ
·
~π
Σ
†
Σ = ΣΣ
†
= (
σ
2
+
~π
2
)
I
h
0

Σ

0
i
=
fI
=
1
4
tr
(
D
μ
Σ
†
D
μ
Σ
)

V
(Σ)

1
4
~
W
μν
L
·
~
W
Lμν

1
4
~
W
μν
R
·
~
W
Rμν
D
μ
Σ =
∂
μ
Σ +
ig
L
~
W
μ
L
·
~
τ
2
Σ

ig
R
~
W
μ
R
·
Σ
~
τ
2
now RH
SU
(2)
is a
gauge symmetry
δ
Σ =
i~
²
L
·
~
τ
2
Σ

i
Σ
~
²
R
·
~
τ
2
~
²
L
=
~
²
R
is unbroken for
h
0

Σ

0
i
=
fI
because
[
~
τ, I
]
= 0
L
(Σ
, W
) =
1
2
‡
∂
μ
σ

i~
τ
·
(
∂
μ
~π
+
g
L
f
~
W
Lμ
/
2

g
R
f
~
W
Rμ
/
2)
·
‡
∂
μ
σ
+
i~
τ
·
(
∂
μ
~π
+
g
L
f
~
W
μ
L
/
2

g
R
f
~
W
μ
R
/
2)
·

λf
2
2
σ
2

1
4
~
W
μν
L
·
~
W
Lμν

1
4
~
W
μν
R
·
~
W
Rμν
scalar with mass
√
λf
vectors with mass ??
six gauge generators are broken  but three combinations are unbroken — so
there were three GB to be eaten to become the longitudinal components of a
triplet of massive gauge bosons and a triplet remains massless
19
SU
(2)
×
SU
(2)
coupled to
σ
model
L
(Σ
, W
) =
1
2
‡
∂
μ
σ

i~
τ
·
(
∂
μ
~π
+
g
L
f
~
W
Lμ
/
2

g
R
f
~
W
Rμ
/
2)
·
‡
∂
μ
σ
+
i~
τ
·
(
∂
μ
~π
+
g
L
f
~
W
μ
L
/
2

g
R
f
~
W
μ
R
/
2)
·

λf
2
2
σ
2

1
4
~
W
μν
L
·
~
W
Lμν

1
4
~
W
μν
R
·
~
W
Rμν
scalar with mass
√
λf
vectors with mass
p
g
2
L
+
g
2
R
f/
2
g
L
~
W
μ
L

g
R
~
W
μ
R
p
g
2
L
+
g
2
R
this is the linear
combination that appears
in the mass term
~
W
μ
L
/g
L
+
~
W
μ
R
/g
R
p
1
/g
2
L
+ 1
/g
2
R
this is the linear
combination that couples to
the unbroken generators
~
W
μ
L
/g
L
+
~
W
μ
R
/g
R
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 Spring '10
 GEORGI
 Quantum Field Theory, ... ..., wA, Quantum chromodynamics, Gauge theory, Dµ Dµ