4 QCD: The Theory of Strong Interactions
87
statistics. In QCD this requirement is very simply satisfied by abc q a q b q c where a,
b, c are SU (3) colour indices. (c) The choice of SU (N C = 3) colour is confirmed by
many processes that directly measure N C . Some examples are listed here. The total
rate for hadronic production in e + e − annihilation is linear in N C . Precisely if we
consider R = σ (e + e − → hadrons)/σ point (e + e − → μ + μ − ) above b ¯
b threshold
and below m Z and we neglect small computable radiative corrections (that will be
discussed later in Sect. 4.5) we have a sum of individual contributions (proportional
to Q 2 , where Q is the electric charge in units of the proton charge) from q ¯
q final
states with q = u, c, d, s, b:
R ≈ N C [2 ·
4
9
+ 3 ·
1
9
] ≈ N C
11
9
(4.7)
The data neatly indicate N C = 3 as seen from Fig. 4.2 [9]. The slight excess of
the data with respect to the value 11/3 is due to the QCD radiative corrections
(Sect. 4.5). Similarly we can consider the branching ratio B(W − → e − ¯
ν), again
in Born approximation. The possible fermion-antifermion (f ¯
f ) final states are for
f = e − , μ − , τ − , d, s (there is no f = b because the top quark is too heavy for b ¯
t
to occur). Each channel gives the same contribution, except that for quarks we have
N C colours:
B(W
−
→ e
−
¯
ν) ≈
1
3 + 2N C
(4.8)
Fig. 4.2 Comparison of the data on R = σ (e + e − → hadrons)/σ point (e + e − → μ + μ − ) with
the QCD prediction [9]. N C = 3 is indicated
87
statistics. In QCD this requirement is very simply satisfied by abc q a q b q c where a,
b, c are SU (3) colour indices. (c) The choice of SU (N C = 3) colour is confirmed by
many processes that directly measure N C . Some examples are listed here. The total
rate for hadronic production in e + e − annihilation is linear in N C . Precisely if we
consider R = σ (e + e − → hadrons)/σ point (e + e − → μ + μ − ) above b ¯
b threshold
and below m Z and we neglect small computable radiative corrections (that will be
discussed later in Sect. 4.5) we have a sum of individual contributions (proportional
to Q 2 , where Q is the electric charge in units of the proton charge) from q ¯
q final
states with q = u, c, d, s, b:
R ≈ N C [2 ·
4
9
+ 3 ·
1
9
] ≈ N C
11
9
(4.7)
The data neatly indicate N C = 3 as seen from Fig. 4.2 [9]. The slight excess of
the data with respect to the value 11/3 is due to the QCD radiative corrections
(Sect. 4.5). Similarly we can consider the branching ratio B(W − → e − ¯
ν), again
in Born approximation. The possible fermion-antifermion (f ¯
f ) final states are for
f = e − , μ − , τ − , d, s (there is no f = b because the top quark is too heavy for b ¯
t
to occur). Each channel gives the same contribution, except that for quarks we have
N C colours:
B(W
−
→ e
−
¯
ν) ≈
1
3 + 2N C
(4.8)
Fig. 4.2 Comparison of the data on R = σ (e + e − → hadrons)/σ point (e + e − → μ + μ − ) with
the QCD prediction [9]. N C = 3 is indicated