80
14. QCD I: Introduction, Tree Graph Predictions, and Jets
where W is a special unitary 3 × 3 matrix parametrized as
W = exp(iα · λ/2),
(14.15)
and ψ
† transforms as
ψ
†
→ ψ
†′ = ψ
† W
† .
(14.16)
The proof of the invariance of ψ
† ψ goes through as in (14.13), and it can be
shown (problem 14.1(a)) that the antisymmetric 3q combination (14.2) is also
an SU(3) c invariant. Thus both the proposed meson and baryon states are
colour singlets. It is not possible to choose the λ’s to be pure imaginary in
(14.15), and thus the 3 × 3 W matrices of SU(3) c cannot be real, so that there
is a distinction between ψ and ψ
∗ , as we learned in section 12.2. Indeed, it
can be shown (see Carruthers 1966, chapter 3, Jones 1990, chapter 8, and also
problem 14.1(b)) that, unlike the case of SU(2) c triplets, it is not possible to
form an SU(3) c colour singlet combination out of two colour triplets qq or
anti-triplets ¯
q¯ q. Thus SU(3) c seems to be a possible and economical choice
for the colour group.
14.2.2
Global SU(3) c invariance, and ‘scalar gluons’
As stated above, we are assuming, on empirical grounds, that the only physically observed hadronic states are colour singlets – and this now means singlets under SU(3) c . What sort of interquark force could produce this dramatic
result? Consider an SU(2) analogy again, the interaction of two nucleons belonging to the lowest (doublet) representation of SU(2). Labelling the states
by an isospin T , the possible T values for two nucleons are T = 1 (triplet) and
T = 0 (singlet). We know of an isospin-dependent force which can produce a
splitting between these states, namely V τ 1 · τ 2 , where the ‘1’ and ‘2’ refer to
1
the two nucleons. The total isospin is T = (τ 1 + τ 2 ), and we have
2
T
2 =
1 (τ 1
2 + 2τ 1 · τ 2 + τ
2
2 ) =
1 (3 + 2τ 1 · τ 2 + 3)
(14.17)
4
4
whence
τ 1 · τ 2 = 2T
2
− 3.
(14.18)
In the triplet state T
2 = 2, and in the singlet state T
2 = 0. Thus
(τ 1 · τ 2 ) T =1 = 1
(14.19)
(τ 1 · τ 2 ) T =0 = −3
(14.20)
and if V is positive the T = 0 state is pulled down. A similar thing happens
in SU(3) c . Suppose this interquark force depended on the quark colours via
a term proportional to
λ 1 · λ 2 .
(14.21)
14. QCD I: Introduction, Tree Graph Predictions, and Jets
where W is a special unitary 3 × 3 matrix parametrized as
W = exp(iα · λ/2),
(14.15)
and ψ
† transforms as
ψ
†
→ ψ
†′ = ψ
† W
† .
(14.16)
The proof of the invariance of ψ
† ψ goes through as in (14.13), and it can be
shown (problem 14.1(a)) that the antisymmetric 3q combination (14.2) is also
an SU(3) c invariant. Thus both the proposed meson and baryon states are
colour singlets. It is not possible to choose the λ’s to be pure imaginary in
(14.15), and thus the 3 × 3 W matrices of SU(3) c cannot be real, so that there
is a distinction between ψ and ψ
∗ , as we learned in section 12.2. Indeed, it
can be shown (see Carruthers 1966, chapter 3, Jones 1990, chapter 8, and also
problem 14.1(b)) that, unlike the case of SU(2) c triplets, it is not possible to
form an SU(3) c colour singlet combination out of two colour triplets qq or
anti-triplets ¯
q¯ q. Thus SU(3) c seems to be a possible and economical choice
for the colour group.
14.2.2
Global SU(3) c invariance, and ‘scalar gluons’
As stated above, we are assuming, on empirical grounds, that the only physically observed hadronic states are colour singlets – and this now means singlets under SU(3) c . What sort of interquark force could produce this dramatic
result? Consider an SU(2) analogy again, the interaction of two nucleons belonging to the lowest (doublet) representation of SU(2). Labelling the states
by an isospin T , the possible T values for two nucleons are T = 1 (triplet) and
T = 0 (singlet). We know of an isospin-dependent force which can produce a
splitting between these states, namely V τ 1 · τ 2 , where the ‘1’ and ‘2’ refer to
1
the two nucleons. The total isospin is T = (τ 1 + τ 2 ), and we have
2
T
2 =
1 (τ 1
2 + 2τ 1 · τ 2 + τ
2
2 ) =
1 (3 + 2τ 1 · τ 2 + 3)
(14.17)
4
4
whence
τ 1 · τ 2 = 2T
2
− 3.
(14.18)
In the triplet state T
2 = 2, and in the singlet state T
2 = 0. Thus
(τ 1 · τ 2 ) T =1 = 1
(14.19)
(τ 1 · τ 2 ) T =0 = −3
(14.20)
and if V is positive the T = 0 state is pulled down. A similar thing happens
in SU(3) c . Suppose this interquark force depended on the quark colours via
a term proportional to
λ 1 · λ 2 .
(14.21)