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Standard Model and Beyond
· Problem 3.8
Gauge Theories
Physics — Particle Physics · som-invariance-lagrangian
Problem
Show that
L
=
1
2
[
(
∂
μ
ϕ
a
)
2
−
μ
2
ϕ
a
2
]
−
λ
(
ϕ
a
2
)
2
,
\mathcal{L} = \frac{1}{2}\left[(\partial_\mu \phi_a)^2 - \mu^2 \phi_a^2\right] - \lambda\left(\phi_a^2\right)^2,
L
=
2
1
[
(
∂
μ
ϕ
a
)
2
−
μ
2
ϕ
a
2
]
−
λ
(
ϕ
a
2
)
2
,
with
ϕ
a
=
ϕ
a
†
\phi_a = \phi_a^\dagger
ϕ
a
=
ϕ
a
†
, is invariant under infinitesimal
S
O
(
m
)
SO(m)
S
O
(
m
)
transformations (i.e.,
[
T
i
,
L
]
=
0
\left[T^i, \mathcal{L}\right] = 0
[
T
i
,
L
]
=
0
), with
ϕ
a
\phi_a
ϕ
a
,
a
=
1
⋯
m
a = 1 \cdots m
a
=
1
⋯
m
transforming as the vector representation.
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