22.2. The SU(2) × U(1) electroweak gauge theory
385
Turning now to the leptonic weak neutral current, this will appear via the
couplings to the Z
0 , written as
− ˆ μ
ˆ
j NC (leptons)Z μ .
(22.57)
ˆ
ˆ
Referring to (22.36) for the linear combination of W
μ
3 and B
μ which represents
ˆ
Z
μ , we find (problem 22.4)
[ (
)
]
g ∑ ¯
l
− γ
ˆ μ
ˆ μ
1
5
2
ˆ
j NC (leptons) =
ψ γ t 3
sin
l ψ
22.
θ
l
θ W Q
l ,
( 58)
cos W
2
l
−
where the sum is over the six lepton fields ν e , e
− , ν μ , . . . τ
− . For the Q = 0
neutrinos with t 3 = +
1 ,
2
∑
ˆ μ
g
)
¯
j NC (neutrinos) =
ν ˆ
μ (1
l
− γ 5
γ
ν ˆ l ,
(22.59)
2 cos θ W
2
l
where now l = e, μ, τ. For the other (negatively charged) leptons, we shall
have both L and R couplings from (22.58), and we can write
[ (
)
(
)]
∑
ˆ μ
g
¯ ˆ μ l 1 − γ 5
1 +
j NC (cha ged leptons) =
lγ c
cos θ
L
+ c
l
γ 5
ˆ
r
W
2
R
l,
2
l=e,μ,τ
(22.60)
where
c
l
L = t
l
3 − sin
2
1
θ W Q l = − + sin
2 θ W
(22.61)
2
c
l
R
= − sin
2 θ W Q l = sin
2 θ W .
(22.62)
As noted earlier, the Z
0 coupling is not pure ‘V–A’. These relations (22.59)–
(22.62) are exactly the ones given earlier, in (20.85)–(20.87); in particular,
the couplings are independent of ‘l’ and hence exhibit lepton universality.
The alternative notation
μ
g
∑ ¯
ˆ
ˆ
j NC (charged leptons) =
lγ
μ (g
l
2 cos θ
V − g
l
ˆ
A γ 5 )l
(22.63)
W l
is often used, where
g
l
1
1
V = − + 2 sin
2 θ W
g
l
A =
2
− , independent of l.
(22.64)
2
Note that the g
l
V vanishes for sin
2 θ W = 0.25. Again, the Feynman rules for
lepton-Z couplings (appendix Q) are contained in (22.59) and (22.60).
As in the case of W-mediated charge-charging processes, Z
0 -mediated processes reduce to the current–current form at low k
2 . For example, the amplitude for e
− μ
−
→ e
− μ
− via Z
0 exchange (figure 22.11) reduces to
ig
2
−
u ¯(e)γ [c
l
μ L (1 − γ 5 ) + c
l (1
μ
4 cos 2 θ W M
R
+ γ 5 )]u(e)u ¯( )γ
μ
2
Z
× [c
l
L (1 − γ 5 ) + c
l
R (1 + γ 5 )]u(μ).
(22.65)
385
Turning now to the leptonic weak neutral current, this will appear via the
couplings to the Z
0 , written as
− ˆ μ
ˆ
j NC (leptons)Z μ .
(22.57)
ˆ
ˆ
Referring to (22.36) for the linear combination of W
μ
3 and B
μ which represents
ˆ
Z
μ , we find (problem 22.4)
[ (
)
]
g ∑ ¯
l
− γ
ˆ μ
ˆ μ
1
5
2
ˆ
j NC (leptons) =
ψ γ t 3
sin
l ψ
22.
θ
l
θ W Q
l ,
( 58)
cos W
2
l
−
where the sum is over the six lepton fields ν e , e
− , ν μ , . . . τ
− . For the Q = 0
neutrinos with t 3 = +
1 ,
2
∑
ˆ μ
g
)
¯
j NC (neutrinos) =
ν ˆ
μ (1
l
− γ 5
γ
ν ˆ l ,
(22.59)
2 cos θ W
2
l
where now l = e, μ, τ. For the other (negatively charged) leptons, we shall
have both L and R couplings from (22.58), and we can write
[ (
)
(
)]
∑
ˆ μ
g
¯ ˆ μ l 1 − γ 5
1 +
j NC (cha ged leptons) =
lγ c
cos θ
L
+ c
l
γ 5
ˆ
r
W
2
R
l,
2
l=e,μ,τ
(22.60)
where
c
l
L = t
l
3 − sin
2
1
θ W Q l = − + sin
2 θ W
(22.61)
2
c
l
R
= − sin
2 θ W Q l = sin
2 θ W .
(22.62)
As noted earlier, the Z
0 coupling is not pure ‘V–A’. These relations (22.59)–
(22.62) are exactly the ones given earlier, in (20.85)–(20.87); in particular,
the couplings are independent of ‘l’ and hence exhibit lepton universality.
The alternative notation
μ
g
∑ ¯
ˆ
ˆ
j NC (charged leptons) =
lγ
μ (g
l
2 cos θ
V − g
l
ˆ
A γ 5 )l
(22.63)
W l
is often used, where
g
l
1
1
V = − + 2 sin
2 θ W
g
l
A =
2
− , independent of l.
(22.64)
2
Note that the g
l
V vanishes for sin
2 θ W = 0.25. Again, the Feynman rules for
lepton-Z couplings (appendix Q) are contained in (22.59) and (22.60).
As in the case of W-mediated charge-charging processes, Z
0 -mediated processes reduce to the current–current form at low k
2 . For example, the amplitude for e
− μ
−
→ e
− μ
− via Z
0 exchange (figure 22.11) reduces to
ig
2
−
u ¯(e)γ [c
l
μ L (1 − γ 5 ) + c
l (1
μ
4 cos 2 θ W M
R
+ γ 5 )]u(e)u ¯( )γ
μ
2
Z
× [c
l
L (1 − γ 5 ) + c
l
R (1 + γ 5 )]u(μ).
(22.65)