74
14. QCD I: Introduction, Tree Graph Predictions, and Jets
which is an essential tool in the modern confrontation of perturbative QCD
with data. Some of the simpler predictions of the renormalization group technique will be compared with experimental data in the last part of chapter
15.
In chapter 16 we work towards understanding some non-perturbative aspects of QCD. As a natural concomitant of asymptotic freedom, it is to be
expected that the effective coupling strength becomes progressively larger at
longer distances or lower energies, ultimately being strong enough to lead
(presumably) to the confinement of quarks and gluons; this is sometimes referred to as ‘infrared slavery’. In this regime perturbation theory clearly fails.
An alternative, purely numerical, approach is available however, namely the
method of ‘lattice’ QCD, which involves replacing the space-time continuum
by a discrete lattice of points. At first sight, this may seem a topic rather
disconnected from everything that has preceded it. But we shall see that in
fact it provides some powerful new insights into several aspects of quantum
field theory in general, and in particular of renormalization, by revisiting it in
coordinate (rather than momentum) space. Quite apart from this, however,
results from lattice QCD now provide independent confirmation of the theory,
in the non-perturbative regime.
14.1 The colour degree of freedom
The first intimation of a new, unrevealed degree of freedom of matter came
from baryon spectroscopy (Greenberg 1964; see also Han and Nambu 1965,
and Tavkhelidze 1965). For a baryon made of three spin1 quarks, the original
2
non-relativistic quark model wave-function took the form
ψ 3q = ψ 3q,space ψ 3q,spin ψ 3q,flavour .
(14.1)
It was soon realized (e.g. Dalitz 1965) that the product of these space, spin
and flavour wavefunctions for the ground state baryons was symmetric under
interchange of any two quarks. For example, the Δ
++ state mentioned in
section 12.2.3 is made of three u quarks (flavour symmetric) in the J
P =
+
3
state, which has zero orbital angular momentum and is hence spatially
2
3
symmetric, and a symmetric S = spin wavefunction. But we saw in section
2
7.2 that quantum field theory requires fermions to obey the exclusion principle
– i.e. the wavefunction ψ 3q should be antisymmetric with respect to quark
interchange. A simple way of implementing this requirement is to suppose
that the quarks carry a further degree of freedom, called colour, with respect
to which the 3q wavefunction can be antisymmetrized, as follows (Fritzsch
and Gell-Mann 1972, Bardeen, Fritzsch and Gell-Mann 1973). We introduce
a colour wavefunction with colour index α:
ψ α (α = 1, 2, 3).
14. QCD I: Introduction, Tree Graph Predictions, and Jets
which is an essential tool in the modern confrontation of perturbative QCD
with data. Some of the simpler predictions of the renormalization group technique will be compared with experimental data in the last part of chapter
15.
In chapter 16 we work towards understanding some non-perturbative aspects of QCD. As a natural concomitant of asymptotic freedom, it is to be
expected that the effective coupling strength becomes progressively larger at
longer distances or lower energies, ultimately being strong enough to lead
(presumably) to the confinement of quarks and gluons; this is sometimes referred to as ‘infrared slavery’. In this regime perturbation theory clearly fails.
An alternative, purely numerical, approach is available however, namely the
method of ‘lattice’ QCD, which involves replacing the space-time continuum
by a discrete lattice of points. At first sight, this may seem a topic rather
disconnected from everything that has preceded it. But we shall see that in
fact it provides some powerful new insights into several aspects of quantum
field theory in general, and in particular of renormalization, by revisiting it in
coordinate (rather than momentum) space. Quite apart from this, however,
results from lattice QCD now provide independent confirmation of the theory,
in the non-perturbative regime.
14.1 The colour degree of freedom
The first intimation of a new, unrevealed degree of freedom of matter came
from baryon spectroscopy (Greenberg 1964; see also Han and Nambu 1965,
and Tavkhelidze 1965). For a baryon made of three spin1 quarks, the original
2
non-relativistic quark model wave-function took the form
ψ 3q = ψ 3q,space ψ 3q,spin ψ 3q,flavour .
(14.1)
It was soon realized (e.g. Dalitz 1965) that the product of these space, spin
and flavour wavefunctions for the ground state baryons was symmetric under
interchange of any two quarks. For example, the Δ
++ state mentioned in
section 12.2.3 is made of three u quarks (flavour symmetric) in the J
P =
+
3
state, which has zero orbital angular momentum and is hence spatially
2
3
symmetric, and a symmetric S = spin wavefunction. But we saw in section
2
7.2 that quantum field theory requires fermions to obey the exclusion principle
– i.e. the wavefunction ψ 3q should be antisymmetric with respect to quark
interchange. A simple way of implementing this requirement is to suppose
that the quarks carry a further degree of freedom, called colour, with respect
to which the 3q wavefunction can be antisymmetrized, as follows (Fritzsch
and Gell-Mann 1972, Bardeen, Fritzsch and Gell-Mann 1973). We introduce
a colour wavefunction with colour index α:
ψ α (α = 1, 2, 3).