14.1. The colour degree of freedom
75
We are here writing the three labels as ‘1, 2, 3’, but they are often referred to
by colour names such as ‘red, blue, green’; it should be understood that this
is merely a picturesque way of referring to the three basic states of this degree
of freedom, and has nothing to do with real colour! With the addition of this
degree of freedom we can certainly form a three-quark wavefunction which is
antisymmetric in colour by using the antisymmetric symbol ∈ αβγ , namely
1
ψ 3q, colour = ∈ αβγ ψ α ψ β ψ γ
(14.2)
and this must then be multiplied into (14.1) to give the full 3q wavefunction.
To date, all known baryon states can be described this way, i.e. the symmetry
of the ‘traditional’ space-spin-flavour wavefunction (14.1) is symmetric overall,
while the required antisymmetry is restored by the additional factor (14.2). As
far as meson (qq) ¯ states are concerned, what was previously a π
+ wavefunction
d
∗ u is now
1
√ (d
∗
1 u 1 + d
∗
2 u 2 + d
∗
3 u 3 )
(14.3)
3
which we write in general as (1/
√
3)d
†
α u α . We shall shortly see the group
theoretical significance of this ‘neutral superposition’, and of (14.2). Meanwhile, we note that (14.2) is actually the only way of making an antisymmetric
combination of the three ψ’s; it is therefore called a (colour) singlet. It is reassuring that there is only one way of doing this – otherwise, we would have
obtained more baryon states than are physically observed. As we shall see in
section 14.2.1, (14.3) is also a colour singlet combination.
The above would seem a somewhat artificial device unless there were some
physical consequences of this increase in the number of quark types – and there
are. In any process which we can describe in terms of creation or annihilation
of quarks, the multiplicity of quark types will enter into the relevant observable
cross section or decay rate. For example, at high energies the ratio
σ(e
+ e
−
→ hadrons)
R =
(14.4)
σ(e + e − → μ + μ − )
will, in the quark parton model (see section 9.5), reflect the magnitudes of the
individual quark couplings to the photon:
∑
R =
e
2
a
(14.5)
a
where a runs over all quark types. For five quarks u, d, s, c, b with respective
charges
2 , −
1 , −
1 ,
2 ,
t
3
3 3 −
1 , his yields
3
3
11
R no colour =
(14.6)
9
1 In (14.2) each ψ refers to a different quark, but we have not indicated the quark labels
explicitly.
75
We are here writing the three labels as ‘1, 2, 3’, but they are often referred to
by colour names such as ‘red, blue, green’; it should be understood that this
is merely a picturesque way of referring to the three basic states of this degree
of freedom, and has nothing to do with real colour! With the addition of this
degree of freedom we can certainly form a three-quark wavefunction which is
antisymmetric in colour by using the antisymmetric symbol ∈ αβγ , namely
1
ψ 3q, colour = ∈ αβγ ψ α ψ β ψ γ
(14.2)
and this must then be multiplied into (14.1) to give the full 3q wavefunction.
To date, all known baryon states can be described this way, i.e. the symmetry
of the ‘traditional’ space-spin-flavour wavefunction (14.1) is symmetric overall,
while the required antisymmetry is restored by the additional factor (14.2). As
far as meson (qq) ¯ states are concerned, what was previously a π
+ wavefunction
d
∗ u is now
1
√ (d
∗
1 u 1 + d
∗
2 u 2 + d
∗
3 u 3 )
(14.3)
3
which we write in general as (1/
√
3)d
†
α u α . We shall shortly see the group
theoretical significance of this ‘neutral superposition’, and of (14.2). Meanwhile, we note that (14.2) is actually the only way of making an antisymmetric
combination of the three ψ’s; it is therefore called a (colour) singlet. It is reassuring that there is only one way of doing this – otherwise, we would have
obtained more baryon states than are physically observed. As we shall see in
section 14.2.1, (14.3) is also a colour singlet combination.
The above would seem a somewhat artificial device unless there were some
physical consequences of this increase in the number of quark types – and there
are. In any process which we can describe in terms of creation or annihilation
of quarks, the multiplicity of quark types will enter into the relevant observable
cross section or decay rate. For example, at high energies the ratio
σ(e
+ e
−
→ hadrons)
R =
(14.4)
σ(e + e − → μ + μ − )
will, in the quark parton model (see section 9.5), reflect the magnitudes of the
individual quark couplings to the photon:
∑
R =
e
2
a
(14.5)
a
where a runs over all quark types. For five quarks u, d, s, c, b with respective
charges
2 , −
1 , −
1 ,
2 ,
t
3
3 3 −
1 , his yields
3
3
11
R no colour =
(14.6)
9
1 In (14.2) each ψ refers to a different quark, but we have not indicated the quark labels
explicitly.