88
G. Altarelli and S. Forte
For N C = 3 we obtain B = 11% and the experimental number is B = 10.7%.
Another analogous example is the branching ratio B(τ − → e − ¯
ν e ν τ ). From the final
state channels with f = e − , μ − , d we find
B(τ
−
→ e
−
¯
ν e ν τ ) ≈
1
2 + N C
(4.9)
For N C = 3 we obtain B = 20% and the experimental number is B = 18% (the less
accuracy in this case is explained by the larger radiative and phase-space corrections
because the mass of τ − is much smaller than m W ). An important process that is
quadratic in N C is the rate (π 0 → 2γ ). This rate can be reliably calculated from a
solid theorem in field theory which has to do with the chiral anomaly:
(π
0
→ 2γ ) ≈ (
N C
3
)
2
α 2 m 3
π 0
32π 3 f 2
π
= (7.73 ± 0.04)(
N C
3
)
2 eV
(4.10)
where the prediction is obtained for f π = (130.7 ± 0.37) MeV. The experimental
result is = (7.7±0.5) eV in remarkable agreement with N C = 3. There are many
more experimental confirmations that N C = 3: for example the rate for Drell-Yan
processes (see Sect. 5.4) is inversely proportional to N C .
How do we get testable predictions from QCD? On the one hand there are non
perturbative methods. The most important at present is the technique of lattice
simulations: it is based on first principles, it has produced very valuable results on
confinement, phase transitions, bound states, hadronic matrix elements and so on,
and it is by now an established basic tool. The main limitation is from computing
power and therefore there is continuous progress and a lot of good perspectives
for the future. Another class of approaches is based on effective lagrangians which
provide simpler approximations than the full theory, valid in some definite domain
of physical conditions. Chiral lagrangians are based on soft pion theorems and are
valid for suitable processes at energies below 1 GeV. Heavy quark effective theories
are obtained from expanding in inverse powers of the heavy quark mass and are
mainly important for the study of b and, to less accuracy, c decays. The approach of
QCD sum rules has led to interesting results but appears to offer not much potential
for further development. Similarly specific potential models for quarkonium have a
limited range of application. On the other hand, the perturbative approach, based on
asymptotic freedom, still remains the main quantitative connection to experiment,
due to its wide range of applicability to all sorts of “hard” processes. To perturbative
QCD will be devoted the next sections.
4.2 Massless QCD and Scale Invariance
As discussed in Chap. 2, the QCD lagrangian in Eq. (4.1) only specifies the theory
at the classical level. The procedure for quantisation of gauge theories involves a
number of complications that arise from the fact that not all degrees of freedom of
G. Altarelli and S. Forte
For N C = 3 we obtain B = 11% and the experimental number is B = 10.7%.
Another analogous example is the branching ratio B(τ − → e − ¯
ν e ν τ ). From the final
state channels with f = e − , μ − , d we find
B(τ
−
→ e
−
¯
ν e ν τ ) ≈
1
2 + N C
(4.9)
For N C = 3 we obtain B = 20% and the experimental number is B = 18% (the less
accuracy in this case is explained by the larger radiative and phase-space corrections
because the mass of τ − is much smaller than m W ). An important process that is
quadratic in N C is the rate (π 0 → 2γ ). This rate can be reliably calculated from a
solid theorem in field theory which has to do with the chiral anomaly:
(π
0
→ 2γ ) ≈ (
N C
3
)
2
α 2 m 3
π 0
32π 3 f 2
π
= (7.73 ± 0.04)(
N C
3
)
2 eV
(4.10)
where the prediction is obtained for f π = (130.7 ± 0.37) MeV. The experimental
result is = (7.7±0.5) eV in remarkable agreement with N C = 3. There are many
more experimental confirmations that N C = 3: for example the rate for Drell-Yan
processes (see Sect. 5.4) is inversely proportional to N C .
How do we get testable predictions from QCD? On the one hand there are non
perturbative methods. The most important at present is the technique of lattice
simulations: it is based on first principles, it has produced very valuable results on
confinement, phase transitions, bound states, hadronic matrix elements and so on,
and it is by now an established basic tool. The main limitation is from computing
power and therefore there is continuous progress and a lot of good perspectives
for the future. Another class of approaches is based on effective lagrangians which
provide simpler approximations than the full theory, valid in some definite domain
of physical conditions. Chiral lagrangians are based on soft pion theorems and are
valid for suitable processes at energies below 1 GeV. Heavy quark effective theories
are obtained from expanding in inverse powers of the heavy quark mass and are
mainly important for the study of b and, to less accuracy, c decays. The approach of
QCD sum rules has led to interesting results but appears to offer not much potential
for further development. Similarly specific potential models for quarkonium have a
limited range of application. On the other hand, the perturbative approach, based on
asymptotic freedom, still remains the main quantitative connection to experiment,
due to its wide range of applicability to all sorts of “hard” processes. To perturbative
QCD will be devoted the next sections.
4.2 Massless QCD and Scale Invariance
As discussed in Chap. 2, the QCD lagrangian in Eq. (4.1) only specifies the theory
at the classical level. The procedure for quantisation of gauge theories involves a
number of complications that arise from the fact that not all degrees of freedom of