242 CHAPTER 4. UNIFIED FIELD THEORY
To match (4.6.40) and (4.6.41), we take theSU( 2 )generator transformation as follows
σ ̃ 1
σ ̃ 2
σ ̃ 3
=√^1
2
1 i 0
1 −i 0
0 0√
2
σ 1
σ 2
σ 3
.
Under this transformation,Wμaandφaare changed into
(W ̃μ^1 ,W ̃μ^2 ,W ̃μ^3 ) = (Wμ±,Zμ) =( 1
√
2
(Wμ^1 ±iWμ^2 ),Wμ^3)
,
(φ ̃^1 ,φ ̃^2 ,φ ̃^3 ) = (φ±,φ^0 ) =( 1
√
2
(φ^1 ±iφ^2 ),φ^3)
.
The equations (4.6.38) and (4.6.39) obey PRI, and under the above transformation they be-
comes
∂νWν μ±±igw
hc ̄(4.6.43) gα β(W±)α μZβ−Zα μWβ±)−gwJμ±
=
[
∂μ+kW^2 Wμ±+k^2 ZZμ−1
4
(mHc
̄h) 2
xμ]
φ±,∂νZν μ−
igw
hc ̄(4.6.44) gα β(Wα μ+Wβ−−Wα μ−Wβ+)−gwJ^0 μ
=
[
∂μ+kW^2 Wμ±+k^2 ZZμ−1
4
(m
Hc
̄h) 2
xμ]
φ^0 ,H±+
(mHc
̄h)
H±−
gw
hc ̄(4.6.45) ∂μ(γ ̃bW ̃μbH±)
=
gw
̄hcgα β∂μ( ̃εbc±W ̃α μbW ̃βc)+gw∂μJ±μ−1
4
(m
Hc
h ̄) 2
xμ∂μH±,H^0 +
(m
Hc
̄h) 2
H^0 −
gw
hc ̄(4.6.46) ∂μ(γ ̃bW ̃μbH^0 )
=
gw
̄hcgα β∂μ( ̃εbc^0 W ̃α μbW ̃βc)+gw∂μJ^0 μ−1
4
(mHc
̄h) 2
xμ∂μH^0 ,whereH±=√^12 (φ^1 ±iφ^2 ),H^0 =φ^3 in (4.6.45) and (4.6.46), and
(4.6.47)
Jμ±=1
√
2
(J^1 μ±iJ^2 μ) the charged current,Jμ^0 =J^3 μ the neutral current,Wν μ±=∂νWμ±−∂μWν±±igw
hc ̄(ZμWν±−ZνWμ±),Zν μ=∂νZμ−∂μZν+
igw
hc ̄(Wμ+Wν−−Wν+Wμ−),and
(4.6.48) k^2 W=
gwγ 1
√
2 hc ̄, k^2 Z=gwγ 3
hc ̄