298
20. Introduction to the Phenomenology of Weak Interactions
after many years of effort; the value obtained was consistent with the Glashow–
Salam–Weinberg theory (see section 22.3), with the parameter sin
2 θ W =
0.29 ± 0.05.
It is interesting that some seemingly rather similar processes are forbidden
H
lep
to occur, to first order in ˆ , for example
wk
−
ν ¯ μ + e
−
→ ν ¯ μ + e .
(20.49)
For reasons which will become clearer in section 20.6, (20.49) is called a ‘neutral current’ process, in contrast to all the others (such as β-decay or μ-decay)
we have discussed so far, which are called ‘charged current’ processes. If the
lepton pairs are arranged so as to have no net lepton number (for example
e
− ν ¯ e , μ
+ ν μ , ν μ ν ¯ μ etc.) then pairs with non-zero charge occur in charged current processes, while those with zero charge participate in neutral current
processes. In the case of (20.48), the leptons can be grouped either as (¯ ν e e
− )
which is charged, or as (¯ ν e ν e ) or (e
+ e
− ) which are neutral. On the other
hand, there is no way of pairing the leptons in (20.49) so as to cancel the lepton number and have non-zero charge. So (20.49) is a purely ‘neutral current’
process, while some ‘neutral current’ contribution could be present in (20.48),
in principle. In 1973 such neutral current processes were discovered (Hasert
et al. 1973), generating a whole new wave of experimental activity. Their existence had, in fact, been predicted in the first version of the Standard Model,
due to Glashow (1961). Today we know that charged current processes are
mediated by the W
± bosons, and the neutral current ones by the Z
0 . We shall
discuss the neutral current couplings in section 20.6.
20.5 Calculation of the cross section for ν μ + e
−
→ μ
− + ν e
After so much qualitative discussion it is time to calculate something. We
choose the process (20.45), sometimes called inverse muon decay, which is a
pure ‘charged current’ process. The amplitude, in the Fermi-like V-A current
theory, is
√
M = −i(G F / 2)¯ u(μ, k
′ )γ μ (1 − γ 5 )u(ν μ , k)¯ u(ν e , p
′ )γ
μ (1 − γ 5 )u(e, p). (20.50)
We shall be interested in energies much greater than any of the leptons, and
so we shall work in the massless limit ; this is mainly for ease of calculation –
the full expressions for non-zero masses can be obtained with more effort.
From the general formula (6.129) for 2 → 2 scattering in the CM system,
we have, neglecting all masses,
dσ
1
=
|M|
2
(20.51)
dΩ
64π 2 s
where |M|
2 is the appropriate spin-averaged matrix element squared, as in
20. Introduction to the Phenomenology of Weak Interactions
after many years of effort; the value obtained was consistent with the Glashow–
Salam–Weinberg theory (see section 22.3), with the parameter sin
2 θ W =
0.29 ± 0.05.
It is interesting that some seemingly rather similar processes are forbidden
H
lep
to occur, to first order in ˆ , for example
wk
−
ν ¯ μ + e
−
→ ν ¯ μ + e .
(20.49)
For reasons which will become clearer in section 20.6, (20.49) is called a ‘neutral current’ process, in contrast to all the others (such as β-decay or μ-decay)
we have discussed so far, which are called ‘charged current’ processes. If the
lepton pairs are arranged so as to have no net lepton number (for example
e
− ν ¯ e , μ
+ ν μ , ν μ ν ¯ μ etc.) then pairs with non-zero charge occur in charged current processes, while those with zero charge participate in neutral current
processes. In the case of (20.48), the leptons can be grouped either as (¯ ν e e
− )
which is charged, or as (¯ ν e ν e ) or (e
+ e
− ) which are neutral. On the other
hand, there is no way of pairing the leptons in (20.49) so as to cancel the lepton number and have non-zero charge. So (20.49) is a purely ‘neutral current’
process, while some ‘neutral current’ contribution could be present in (20.48),
in principle. In 1973 such neutral current processes were discovered (Hasert
et al. 1973), generating a whole new wave of experimental activity. Their existence had, in fact, been predicted in the first version of the Standard Model,
due to Glashow (1961). Today we know that charged current processes are
mediated by the W
± bosons, and the neutral current ones by the Z
0 . We shall
discuss the neutral current couplings in section 20.6.
20.5 Calculation of the cross section for ν μ + e
−
→ μ
− + ν e
After so much qualitative discussion it is time to calculate something. We
choose the process (20.45), sometimes called inverse muon decay, which is a
pure ‘charged current’ process. The amplitude, in the Fermi-like V-A current
theory, is
√
M = −i(G F / 2)¯ u(μ, k
′ )γ μ (1 − γ 5 )u(ν μ , k)¯ u(ν e , p
′ )γ
μ (1 − γ 5 )u(e, p). (20.50)
We shall be interested in energies much greater than any of the leptons, and
so we shall work in the massless limit ; this is mainly for ease of calculation –
the full expressions for non-zero masses can be obtained with more effort.
From the general formula (6.129) for 2 → 2 scattering in the CM system,
we have, neglecting all masses,
dσ
1
=
|M|
2
(20.51)
dΩ
64π 2 s
where |M|
2 is the appropriate spin-averaged matrix element squared, as in