Physical Foundations of Cosmology

4.3 Electroweak theory 155

written as

φ=χ

(

ζ 1

ζ 2

)

=χ

(

ζ 2 ∗ ζ 1

−ζ 1 ∗ ζ 2

)(

0

1

)

≡χζφ 0 , (4.47)

whereχis a real field andζ 1 ,ζ 2 are two complex scalar fields satisfying the condition
|ζ 1 |^2 +|ζ 2 |^2 = 1 .The definition of theSU( 2 )matrixζand the constant vectorφ 0
can easily be read off the last equality. Substitutingφ=χζφ 0 in (4.45), we obtain

Lφ=

1

2

∂μχ∂μχ−V(χ^2 )+

χ^2

2

φ† 0

(

gGμ−

1

2

g′Bμ

)(

gGμ−

1

2

g′Bμ

)

φ 0 ,

(4.48)

where

Gμ≡ζ−^1 Aμζ−

i

g

ζ−^1 ∂μζ (4.49)

areSU( 2 )gauge-invariant variables.

Problem 4.11Consider theSU( 2 )transformation
(
ζ 1
ζ 2

)

→

( ̃

ζ 1

ζ ̃ 2

)

=U

(

ζ 1

ζ 2

)

(4.50)

accompanied by theU( 1 )transformationζ ̃→e
2 ig′λ ̃
ζand verify that

ζ→

(

e−

2 ig′λ ̃

ζ 2 ∗ e

2 ig′λ ̃

ζ 1

−e−

2 ig′λ ̃

ζ 1 ∗ e

2 ig′λ ̃

ζ 2

)

=UζE, (4.51)

where

ζ≡

(

ζ 2 ∗ ζ 1

−ζ 1 ∗ ζ 2

)

, E≡

(

e−

i 2 g′λ

0

0 e

i 2 g′λ

)

(4.52)

(HintNote that an arbitrarySU( 2 )matrix has the same form as matrixζwithζ 1 ,ζ 2
replaced by some complex numbersα, β.)

Using this result, it is easy to see that the fieldGμisSU( 2 )gauge-invariant, that
is,

Gμ→Gμ (4.53)

as

ζ→Uζ, Aμ→UAμU−^1 +(i/g)(∂μU)U−^1.

Thus, we have rewritten our original Lagrangian (4.45) in terms ofSU( 2 )gauge-
invariant variablesχ,Gμ,Bμ.