303
20.6. Leptonic weak neutral currents
¯ ˆ l . . . ˆ l. We shall assume the following form for these currents (with one eye on
the GSW theory to come):
(1) neutrino neutral current
(
)
1 − γ 5
ν l ν ¯ l γ
μ
g N c ˆ
ν ˆ l
l = e, μ, τ;
(20.85)
2
(2) charged lepton neutral current
[
]
¯
(1 − γ 5 )
(1 + γ 5 )
ˆ
l
l
ˆ
g N lγ
μ c
+ c R
l
l = e, μ, τ.
(20.86)
L
2
2
This is, of course, by no means the most general possible parametrization.
The neutrino coupling is retained as pure ‘V-A’, while the coupling in the
charged lepton sector is now a combination of ‘V-A’ and ‘V+A’ with certain
l
l
coefficients c and c . We may also write the coupling in terms of ‘V’ and
L
R
l
l
l
l
l
l
‘A’ coefficients defined by c V = c L + c R , c = c L − c . An overall factor g N
A
R
determines the strength of the neutral currents as compared to the charged
ones; the c’s determine the relative amplitudes of the various neutral current
processes.
As we shall see, an essential feature of the GSW theory is its prediction
of weak neutral current processes, with couplings determined in terms of one
parameter of the theory called ‘θ W ’, the ‘weak mixing angle’ (Glashow 1961,
Weinberg 1967). The GSW predictions for the parameter g N and the c’s are
(see equations (22.59)–(22.62))
l
l
g N = g/ cos θ W , c
ν l =
1 , c = −
1 + a, c = a
(20.87)
L
R
2
2
for l = e, μ, τ, where a = sin
2 θ W and g is the SU(2) gauge coupling. Note
that a strong form of ‘universality’ is involved here too: the coefficients are
independent of the ‘flavour’ e, μ or τ , for both neutrinos and charged leptons.
The following reactions are available for experimental measurement (in
addition to the charged current process (20.45) already discussed):
−
−
−
ν μ e
→ ν μ e , ν ¯ μ e
−
→ ν ¯ μ e (NC)
(20.88)
−
−
−
ν e e
→ ν e e , ν ¯ e e
−
→ ν ¯ e e (NC + CC)
(20.89)
where ‘NC’ means neutral current and ‘CC’ charged current. Formulae for
these cross sections are given in section 22.3. The experiments are discussed
and reviewed in Commins and Bucksbaum (1983), Renton (1990), and by
Winter (2000). All observations are in excellent agreement with the GSW
predictions, with θ W determined as sin
2 θ W ⋍ 0.23. The reader must note,
however, that modern precision measurements are sensitive to higher-order
(loop) corrections, which must be included in comparing the full GSW theory
20.6. Leptonic weak neutral currents
¯ ˆ l . . . ˆ l. We shall assume the following form for these currents (with one eye on
the GSW theory to come):
(1) neutrino neutral current
(
)
1 − γ 5
ν l ν ¯ l γ
μ
g N c ˆ
ν ˆ l
l = e, μ, τ;
(20.85)
2
(2) charged lepton neutral current
[
]
¯
(1 − γ 5 )
(1 + γ 5 )
ˆ
l
l
ˆ
g N lγ
μ c
+ c R
l
l = e, μ, τ.
(20.86)
L
2
2
This is, of course, by no means the most general possible parametrization.
The neutrino coupling is retained as pure ‘V-A’, while the coupling in the
charged lepton sector is now a combination of ‘V-A’ and ‘V+A’ with certain
l
l
coefficients c and c . We may also write the coupling in terms of ‘V’ and
L
R
l
l
l
l
l
l
‘A’ coefficients defined by c V = c L + c R , c = c L − c . An overall factor g N
A
R
determines the strength of the neutral currents as compared to the charged
ones; the c’s determine the relative amplitudes of the various neutral current
processes.
As we shall see, an essential feature of the GSW theory is its prediction
of weak neutral current processes, with couplings determined in terms of one
parameter of the theory called ‘θ W ’, the ‘weak mixing angle’ (Glashow 1961,
Weinberg 1967). The GSW predictions for the parameter g N and the c’s are
(see equations (22.59)–(22.62))
l
l
g N = g/ cos θ W , c
ν l =
1 , c = −
1 + a, c = a
(20.87)
L
R
2
2
for l = e, μ, τ, where a = sin
2 θ W and g is the SU(2) gauge coupling. Note
that a strong form of ‘universality’ is involved here too: the coefficients are
independent of the ‘flavour’ e, μ or τ , for both neutrinos and charged leptons.
The following reactions are available for experimental measurement (in
addition to the charged current process (20.45) already discussed):
−
−
−
ν μ e
→ ν μ e , ν ¯ μ e
−
→ ν ¯ μ e (NC)
(20.88)
−
−
−
ν e e
→ ν e e , ν ¯ e e
−
→ ν ¯ e e (NC + CC)
(20.89)
where ‘NC’ means neutral current and ‘CC’ charged current. Formulae for
these cross sections are given in section 22.3. The experiments are discussed
and reviewed in Commins and Bucksbaum (1983), Renton (1990), and by
Winter (2000). All observations are in excellent agreement with the GSW
predictions, with θ W determined as sin
2 θ W ⋍ 0.23. The reader must note,
however, that modern precision measurements are sensitive to higher-order
(loop) corrections, which must be included in comparing the full GSW theory