78
14. QCD I: Introduction, Tree Graph Predictions, and Jets
14.2
The dynamics of colour
14.2.1
Colour as an SU(3) group
We now want to consider the possible dynamical role of colour – in other
words, the way in which the forces between quarks depend on their colours.
We have seen that we seem to need three different quark types for each given
flavour. They must all have the same mass, or else we would observe some
‘fine structure’ in the hadronic levels. Furthermore, and for the same reason,
‘colour’ must be an exact symmetry of the Hamiltonian governing the quark
dynamics. What symmetry group is involved? We shall consider how some
empirical facts suggest that the answer is SU(3) c .
To begin with, it is certainly clear that the interquark force must depend
on colour, since we do not observe ‘colour multiplicity’ of hadronic states: for
∗
∗
example we do not see eight other coloured π
+ ’s (d 1 u 2 , d 3 u 1 , . . . ) degenerate
with the one ‘colourless’ physical π
+ whose wavefunction was given previously. The observed hadronic states are all colour singlets, and the force must
somehow be responsible for this. More particularly, the force has to produce
only those very restricted types of quark configuration which are observed in
the hadron spectrum. Consider again the isospin multiplets in nuclear physics
discussed in section 12.1.2. There is one very striking difference in the par1
ticle physics case: for mesons only T = 0, and 1 occur, and for baryons
2
1
only T = 0, 2 , 1 and
3 , while in nuclei there is nothing in principle to stop
2
5
us finding T = , 3, . . . states. (In fact such nuclear states are hard to iden2
tify experimentally, because they occur at high excitation energy for some of
the isobars – cf figure 1.8(c) – where the levels are very dense). The same
restriction holds for SU(3) f also – only 1’s and 8’s occur for mesons; and only
1’s, 8’s and 10’s for baryons. In quark terms, this of course is what is translated into the recipe: ‘mesons are ¯
qq, baryons are qqq’. It is as if we said,
in nuclear physics, that only A = 2 and A = 3 nuclei exist! Thus the quark
forces must have a dramatic saturation property: apparently no qqq, no qqqq,
¯
qqqqq, . . . states exist. Furthermore, no qq or ¯q states exist either – nor, for
q¯
that matter, do single q’s or ¯
q’s. All this can be summarized by saying that
the quark colour degree of freedom must be confined, a property we shall now
assume and return to in chapter 16.
If we assume that only colour singlet states exist (Fritzsch and Gell-Mann
1972, Bardeen, Fritzsch and Gell-Mann 1973), and that the strong interquark
force depends only on
¯
are
but qq and q¯
colour, the fact that qq states
seen
¯q are
not gives us an important clue as to what group to associate with colour. One
simple possibility might be that the three colours correspond to the components of an SU(2) c triplet ‘ψ’. The antisymmetric, colour singlet, three-quark
baryon wavefunction of (14.2) is then just the triple scalar product ψ 1 ·ψ 2 ×ψ 3 ,
which seems satisfactory. But what about the meson wavefunction? Mesons
are formed of quarks and antiquarks, and we recall from sections 12.1.3 and
14. QCD I: Introduction, Tree Graph Predictions, and Jets
14.2
The dynamics of colour
14.2.1
Colour as an SU(3) group
We now want to consider the possible dynamical role of colour – in other
words, the way in which the forces between quarks depend on their colours.
We have seen that we seem to need three different quark types for each given
flavour. They must all have the same mass, or else we would observe some
‘fine structure’ in the hadronic levels. Furthermore, and for the same reason,
‘colour’ must be an exact symmetry of the Hamiltonian governing the quark
dynamics. What symmetry group is involved? We shall consider how some
empirical facts suggest that the answer is SU(3) c .
To begin with, it is certainly clear that the interquark force must depend
on colour, since we do not observe ‘colour multiplicity’ of hadronic states: for
∗
∗
example we do not see eight other coloured π
+ ’s (d 1 u 2 , d 3 u 1 , . . . ) degenerate
with the one ‘colourless’ physical π
+ whose wavefunction was given previously. The observed hadronic states are all colour singlets, and the force must
somehow be responsible for this. More particularly, the force has to produce
only those very restricted types of quark configuration which are observed in
the hadron spectrum. Consider again the isospin multiplets in nuclear physics
discussed in section 12.1.2. There is one very striking difference in the par1
ticle physics case: for mesons only T = 0, and 1 occur, and for baryons
2
1
only T = 0, 2 , 1 and
3 , while in nuclei there is nothing in principle to stop
2
5
us finding T = , 3, . . . states. (In fact such nuclear states are hard to iden2
tify experimentally, because they occur at high excitation energy for some of
the isobars – cf figure 1.8(c) – where the levels are very dense). The same
restriction holds for SU(3) f also – only 1’s and 8’s occur for mesons; and only
1’s, 8’s and 10’s for baryons. In quark terms, this of course is what is translated into the recipe: ‘mesons are ¯
qq, baryons are qqq’. It is as if we said,
in nuclear physics, that only A = 2 and A = 3 nuclei exist! Thus the quark
forces must have a dramatic saturation property: apparently no qqq, no qqqq,
¯
qqqqq, . . . states exist. Furthermore, no qq or ¯q states exist either – nor, for
q¯
that matter, do single q’s or ¯
q’s. All this can be summarized by saying that
the quark colour degree of freedom must be confined, a property we shall now
assume and return to in chapter 16.
If we assume that only colour singlet states exist (Fritzsch and Gell-Mann
1972, Bardeen, Fritzsch and Gell-Mann 1973), and that the strong interquark
force depends only on
¯
are
but qq and q¯
colour, the fact that qq states
seen
¯q are
not gives us an important clue as to what group to associate with colour. One
simple possibility might be that the three colours correspond to the components of an SU(2) c triplet ‘ψ’. The antisymmetric, colour singlet, three-quark
baryon wavefunction of (14.2) is then just the triple scalar product ψ 1 ·ψ 2 ×ψ 3 ,
which seems satisfactory. But what about the meson wavefunction? Mesons
are formed of quarks and antiquarks, and we recall from sections 12.1.3 and