304
20. Introduction to the Phenomenology of Weak Interactions
with experiment (see section 22.6). The simultaneous fit of data from all four
reactions in terms of the single parameter θ W provides already strong confirmation of the theory – and indeed such confirmation was already emerging
in the late 1970’s and early 1980’s, before the actual discovery of the W
±
and Z
0 bosons. It is also interesting to note that the presence of vector (V)
interactions in the neutral current processes may suggest the possibility of
some kind of link with electromagnetic interactions, which are of course also
‘neutral’ (in this sense) and vector-like. In the GSW theory, this linkage is
provided essentially through the parameter θ W , as we shall see.
20.7 Quark weak currents
We now turn our attention to the weak interactions of quarks. We shall begin
by considering an earlier world, when only two generations (four flavours)
were known.
20.7.1 Two generations
The original version of V-A theory was framed in terms of a nucleonic current
¯
of the form ψ ˆ γ
μ (1 − rγ 5 )ψ ˆ n . With the acceptance of quark substructure it
p
was natural to re-interpret such a hadronic transition by a charged current of
¯
the form ˆ
d,
uγ
μ (1−γ 5 ) ˆ very similar to the charged lepton currents; indeed, here
was a further example of ‘universality’, this time between quarks and leptons.
Detailed comparison with experiment showed, however, that such d → u
transitions were very slightly weaker than the analogous leptonic ones; this
could be established by comparing the rates for n → pe
− ν ¯ e and ¯
μ → ν μ e
− ν ¯ e .
But for quarks (or their hadronic composites) there is a further complication, which is the very familiar phenomenon of flavour change in weak hadronic
processes (recall the discussion in section 1.2.2). The first step towards the
modern theory of quark currents was taken by Cabibbo (1963); in a sense, it
restored universality. Cabibbo postulated that the strength of the hadronic
weak interaction was shared between the ΔS = 0 and ΔS = 1 transitions
(where S is the strangeness quantum number), the latter being relatively
suppressed as compared to the former. According to Cabibbo’s hypothesis,
phrased in terms of quarks, the total weak charged current for u, d and s
quarks is
(1 − γ 5 )
(1 − γ 5 )
μ
ˆ
¯
ˆ
¯
j (u, d, s) = cos θ C uγ
μ
d + sin θ C ˆ
s, ˆ
(20.90)
ˆ
uγ
μ
Cab
2
2
where θ C is the ‘Cabibbo angle’ (not to be confused with θ W ). We can now
postulate a total weak charged current
μ
μ
μ
ˆ j (total) = ˆ j (leptons) + ˆ j (u, d, s),
(20.91)
CC
CC
Cab
20. Introduction to the Phenomenology of Weak Interactions
with experiment (see section 22.6). The simultaneous fit of data from all four
reactions in terms of the single parameter θ W provides already strong confirmation of the theory – and indeed such confirmation was already emerging
in the late 1970’s and early 1980’s, before the actual discovery of the W
±
and Z
0 bosons. It is also interesting to note that the presence of vector (V)
interactions in the neutral current processes may suggest the possibility of
some kind of link with electromagnetic interactions, which are of course also
‘neutral’ (in this sense) and vector-like. In the GSW theory, this linkage is
provided essentially through the parameter θ W , as we shall see.
20.7 Quark weak currents
We now turn our attention to the weak interactions of quarks. We shall begin
by considering an earlier world, when only two generations (four flavours)
were known.
20.7.1 Two generations
The original version of V-A theory was framed in terms of a nucleonic current
¯
of the form ψ ˆ γ
μ (1 − rγ 5 )ψ ˆ n . With the acceptance of quark substructure it
p
was natural to re-interpret such a hadronic transition by a charged current of
¯
the form ˆ
d,
uγ
μ (1−γ 5 ) ˆ very similar to the charged lepton currents; indeed, here
was a further example of ‘universality’, this time between quarks and leptons.
Detailed comparison with experiment showed, however, that such d → u
transitions were very slightly weaker than the analogous leptonic ones; this
could be established by comparing the rates for n → pe
− ν ¯ e and ¯
μ → ν μ e
− ν ¯ e .
But for quarks (or their hadronic composites) there is a further complication, which is the very familiar phenomenon of flavour change in weak hadronic
processes (recall the discussion in section 1.2.2). The first step towards the
modern theory of quark currents was taken by Cabibbo (1963); in a sense, it
restored universality. Cabibbo postulated that the strength of the hadronic
weak interaction was shared between the ΔS = 0 and ΔS = 1 transitions
(where S is the strangeness quantum number), the latter being relatively
suppressed as compared to the former. According to Cabibbo’s hypothesis,
phrased in terms of quarks, the total weak charged current for u, d and s
quarks is
(1 − γ 5 )
(1 − γ 5 )
μ
ˆ
¯
ˆ
¯
j (u, d, s) = cos θ C uγ
μ
d + sin θ C ˆ
s, ˆ
(20.90)
ˆ
uγ
μ
Cab
2
2
where θ C is the ‘Cabibbo angle’ (not to be confused with θ W ). We can now
postulate a total weak charged current
μ
μ
μ
ˆ j (total) = ˆ j (leptons) + ˆ j (u, d, s),
(20.91)
CC
CC
Cab