12.4. Non-Abelian global symmetries in Lagrangian quantum field theory 31
SU(3)-invariant interactions can also be formed. A particularly important one is the ‘SU(3) dot-product’ of two octets (the analogues of the SU(2)
triplets), which arises in the quark-gluon vertex of QCD (see chapters 13 and
14):
∑
λ
−
¯
ig s
q ˆ f γ
μ a
ˆ
q ˆ f A
a
μ .
(12.136)
2
f
In (12.136), q ˆ f stands for the SU(3) c colour triplet
(
)
ˆ
f r
q ˆ
ˆ
f = ( f )
b
(12.137)
ˆ
f g
ˆ
ˆ
ˆ
ˆ
where ‘f ’ is any of the six quark flavour fields u, ˆ d, c, ˆ s, ˆ t, ˆ b, and A
a
μ are the
8 (a = 1, 2, . . . 8) gluon fields. Once again, (12.136) has the form ‘symmetry
current · gauge field’ characteristic of all gauge interactions.
12.4.2 Chiral symmetry
As our final example of a global non-Abelian symmetry, we shall introduce
the idea of chiral symmetry, which is an exact symmetry for fermions in the
limit in which their masses may be neglected. We have seen that the u and
d quarks have indeed very small masses (≤ 5 MeV) on hadronic scales, and
even the s quark mass (∼ 100 MeV) is relatively small. Thus we may certainly
expect some physical signs of the symmetry associated with m u ≈ m d ≈ 0,
and possibly also of the larger symmetry holding when m u ≈ m d ≈ m s ≈ 0.
As we shall see, however, this expectation leads to a puzzle, the resolution of
which will have to be postponed until the concept of ‘spontaneous symmetry
breaking’ has been developed in Part VII.
We begin with the simplest case of just one fermion. Since we are interested
in the ‘small mass’ regime, it is sensible to use the representation (3.40) of
the Dirac matrices, in which the momentum part of the Dirac Hamiltonian is
‘diagonal’ and the mass appears as an ‘off-diagonal’ coupling:
(
)
(
)
σ
0
0 1
α =
, β =
.
(12.138)
0 −σ
1 0
Writing the general Dirac spinor ω as
(
)
φ
ω =
,
(12.139)
χ
we have (as in (4.14), (4.15))
Eφ = σ · pφ + mχ
(12.140)
Eχ = −σ · pχ + mφ.
(12.141)
SU(3)-invariant interactions can also be formed. A particularly important one is the ‘SU(3) dot-product’ of two octets (the analogues of the SU(2)
triplets), which arises in the quark-gluon vertex of QCD (see chapters 13 and
14):
∑
λ
−
¯
ig s
q ˆ f γ
μ a
ˆ
q ˆ f A
a
μ .
(12.136)
2
f
In (12.136), q ˆ f stands for the SU(3) c colour triplet
(
)
ˆ
f r
q ˆ
ˆ
f = ( f )
b
(12.137)
ˆ
f g
ˆ
ˆ
ˆ
ˆ
where ‘f ’ is any of the six quark flavour fields u, ˆ d, c, ˆ s, ˆ t, ˆ b, and A
a
μ are the
8 (a = 1, 2, . . . 8) gluon fields. Once again, (12.136) has the form ‘symmetry
current · gauge field’ characteristic of all gauge interactions.
12.4.2 Chiral symmetry
As our final example of a global non-Abelian symmetry, we shall introduce
the idea of chiral symmetry, which is an exact symmetry for fermions in the
limit in which their masses may be neglected. We have seen that the u and
d quarks have indeed very small masses (≤ 5 MeV) on hadronic scales, and
even the s quark mass (∼ 100 MeV) is relatively small. Thus we may certainly
expect some physical signs of the symmetry associated with m u ≈ m d ≈ 0,
and possibly also of the larger symmetry holding when m u ≈ m d ≈ m s ≈ 0.
As we shall see, however, this expectation leads to a puzzle, the resolution of
which will have to be postponed until the concept of ‘spontaneous symmetry
breaking’ has been developed in Part VII.
We begin with the simplest case of just one fermion. Since we are interested
in the ‘small mass’ regime, it is sensible to use the representation (3.40) of
the Dirac matrices, in which the momentum part of the Dirac Hamiltonian is
‘diagonal’ and the mass appears as an ‘off-diagonal’ coupling:
(
)
(
)
σ
0
0 1
α =
, β =
.
(12.138)
0 −σ
1 0
Writing the general Dirac spinor ω as
(
)
φ
ω =
,
(12.139)
χ
we have (as in (4.14), (4.15))
Eφ = σ · pφ + mχ
(12.140)
Eχ = −σ · pχ + mφ.
(12.141)