14.2. The dynamics of colour
81
Then, in just the same way, we can introduce the total colour operator
1
F = (λ 1 + λ 2 ),
(14.22)
2
so that
F
2
1
= (λ
2 + 2λ · λ + λ
2
(
4
1
1
2
2 )
14.23)
and
λ 1 · λ 2 = 2F
2
− λ
2 ,
(14.24)
where λ
2
1 = λ
2
8
2 = λ
2 , say. Here λ
2
≡ a=1 (λ a )
2 is found (see (12.75)) to
have the value 16/3 (the unit matrix being understood). The operator F
2
commutes with all components of λ 1 and λ 2 (as T
2 does with τ 1 and τ 2 )
ˆ
and represents the quadratic Casimir operator C 2 of SU(3) c (see section M.5
of appendix M), in the colour space of the two quarks considered here. The
ˆ
eigenvalues of C 2 play a very important role in SU(3) c , analogous to that of the
total spin/angular momentum in SU(2). They depend on the SU(3) c representation: indeed, they are one of the defining labels of SU(3) representations
in general (see section M.5). Two quarks, each in the representation 3 c , combine to give a 6 c -dimensional representation and a 3
∗
c (see problem 14.1(b),
ˆ
and Jones (1990) chapter 8). The value of C 2 for the singlet 6 c representation
is 10/3, and for the 3
∗
c representation is 4/3. Thus the ‘λ 1 · λ 2 ’ interaction
will produce a negative (attractive) eigenvalue -8/3 in the 3
∗
c states, but a
repulsive eigenvalue +4/3 in the 6 c states, for two quarks.
The maximum attraction will clearly be for states in which F
2 is zero.
This is the singlet representation 1 c . Two quarks cannot combine to give
a colour singlet state, but we have seen in section 12.2 that a quark and an
antiquark can: they combine to give 1 c and 8 c . In this case (14.24) is replaced
by
1
λ
2
2
2
1 · λ 2 = 2F − (λ
2
1 + λ 2 ),
(14.25)
where ‘1’ refers to the quark and ‘2’ to the antiquark. Thus the ‘λ 1 · λ 2 ’
interaction will give a repulsive eigenvalue +2/3 in the 8 c channel, for which
ˆ
C 2 = 3, and a ‘maximally attractive’ eigenvalue -16/3 in the 1 c channel, for
a quark and an antiquark.
In the case of baryons, built from three quarks, we have seen that when
two of them are coupled to the 3
∗
c state, the eigenvalue of λ 1 · λ 2 is -8/3, one
half of the attraction in the qq ¯ colour singlet state, but still strongly attractive.
The (qq) pair in the 3
∗
c state can then couple to the remaining third quark to
make the overall colour singlet state (14.2), with maximum binding.
Of course, such a simple potential model does not imply that the energy
difference between the 1 c states and all coloured states is infinite, as our
strict ‘colour singlets only’ hypothesis would demand, and which would be
one (rather crude) way of interpreting confinement. Nevertheless, we can ask:
what single particle exchange process between quark (or antiquark) colour
triplets produces a λ 1 · λ 2 type of term? The answer is the exchange of
81
Then, in just the same way, we can introduce the total colour operator
1
F = (λ 1 + λ 2 ),
(14.22)
2
so that
F
2
1
= (λ
2 + 2λ · λ + λ
2
(
4
1
1
2
2 )
14.23)
and
λ 1 · λ 2 = 2F
2
− λ
2 ,
(14.24)
where λ
2
1 = λ
2
8
2 = λ
2 , say. Here λ
2
≡ a=1 (λ a )
2 is found (see (12.75)) to
have the value 16/3 (the unit matrix being understood). The operator F
2
commutes with all components of λ 1 and λ 2 (as T
2 does with τ 1 and τ 2 )
ˆ
and represents the quadratic Casimir operator C 2 of SU(3) c (see section M.5
of appendix M), in the colour space of the two quarks considered here. The
ˆ
eigenvalues of C 2 play a very important role in SU(3) c , analogous to that of the
total spin/angular momentum in SU(2). They depend on the SU(3) c representation: indeed, they are one of the defining labels of SU(3) representations
in general (see section M.5). Two quarks, each in the representation 3 c , combine to give a 6 c -dimensional representation and a 3
∗
c (see problem 14.1(b),
ˆ
and Jones (1990) chapter 8). The value of C 2 for the singlet 6 c representation
is 10/3, and for the 3
∗
c representation is 4/3. Thus the ‘λ 1 · λ 2 ’ interaction
will produce a negative (attractive) eigenvalue -8/3 in the 3
∗
c states, but a
repulsive eigenvalue +4/3 in the 6 c states, for two quarks.
The maximum attraction will clearly be for states in which F
2 is zero.
This is the singlet representation 1 c . Two quarks cannot combine to give
a colour singlet state, but we have seen in section 12.2 that a quark and an
antiquark can: they combine to give 1 c and 8 c . In this case (14.24) is replaced
by
1
λ
2
2
2
1 · λ 2 = 2F − (λ
2
1 + λ 2 ),
(14.25)
where ‘1’ refers to the quark and ‘2’ to the antiquark. Thus the ‘λ 1 · λ 2 ’
interaction will give a repulsive eigenvalue +2/3 in the 8 c channel, for which
ˆ
C 2 = 3, and a ‘maximally attractive’ eigenvalue -16/3 in the 1 c channel, for
a quark and an antiquark.
In the case of baryons, built from three quarks, we have seen that when
two of them are coupled to the 3
∗
c state, the eigenvalue of λ 1 · λ 2 is -8/3, one
half of the attraction in the qq ¯ colour singlet state, but still strongly attractive.
The (qq) pair in the 3
∗
c state can then couple to the remaining third quark to
make the overall colour singlet state (14.2), with maximum binding.
Of course, such a simple potential model does not imply that the energy
difference between the 1 c states and all coloured states is infinite, as our
strict ‘colour singlets only’ hypothesis would demand, and which would be
one (rather crude) way of interpreting confinement. Nevertheless, we can ask:
what single particle exchange process between quark (or antiquark) colour
triplets produces a λ 1 · λ 2 type of term? The answer is the exchange of