36
12. Global Non-Abelian Symmetries
which are axial vectors, and three associated ‘charge’ operators
ˆ
(
1
∫
)
T
2
τ
5
= q ˆ
† γ 5 q ˆd
3
x
(12.166)
2
which are pseudoscalars, belonging to the T=1 representation of SU(2). We
have a new non-Abelian global symmetry, called chiral SU(2) f , which we shall
denote by SU(2) f 5 . As far as their action in the isospinor u-d space is concerned, these chiral charges have exactly the same effect as the ordinary flavour
isospin operators of (12.109). But they are pseudoscalars rather than scalars,
and hence they flip the parity of a state on which they act. Thus, whereas
ˆ (
1 )
the isospin raising operator T
2
+
is such that
ˆ (
1 )
T
2
+ |d› = |u›,
(12.167)
ˆ (
1 )
T
2
+ 5 will also produce a u-type state from a d-type one via
ˆ (
1 )
T
2
+ 5 |d› = |u ˜›,
(12.168)
| ›
| ›
ˆ (
1 ) ˆ
but the u ˜ state will have opposite parity from u . Further, since [T
2
+ 5 , H] =
˜
0, this state u ˜ will be degenerate with d . Similarly, the state d produced
ˆ (
1 )
2
| ›
| ›
| ›
via T 5 |u› will have opposite parity from
−
|d›, and will be degenerate with
|u›. The upshot is that we have two massless states u , d of (say) positive
˜
| › | ›
parity, and a further two massless states |u ˜›, |d› of negative parity, in this
simple model.
Suppose we now let the quarks interact, for example by an interaction of
the QCD type, already indicated in (12.136). In that case, the interaction
terms have the form
λ
¯
uγ ˆ
μ a
¯
ˆ
ˆ
ˆA
a
λ
ˆ
a ˆ
u μ + dγ
μ
dA
a
2
2
μ
(12.169)
where
(
)
(
)
ˆ
u ˆ r
d r
u ˆ = ( u ˆ ) ˆ
, d = ( ˆ
b
d )
b
(12.170)
u ˆ g
ˆ
d g
and the 3 × 3 λ’s act in the r-b-g space. Just as in the previous U(1) case,
the interaction (12.169) is invariant under the global SU(2) f 5 chiral symmetry
(12.164), acting in the u-d space. Note that, somewhat confusingly, (12.169) is
not a simple ‘gauging’ of (12.163): a covariant derivative is being introduced,
but in the space of a new (colour) degree of freedom, not in flavour space. In
fact, the flavour degrees of freedom are ‘inert’ in (12.169), so that it is invariant
under SU(2) f transformations, while the Dirac structure implies that it is also
invariant under chiral SU(2) f 5 transformations (12.164). All the foregoing can
be extended unchanged to chiral SU(3) f 5 , given that QCD is ‘flavour blind’,
and supposing that m s ≈ 0.
12. Global Non-Abelian Symmetries
which are axial vectors, and three associated ‘charge’ operators
ˆ
(
1
∫
)
T
2
τ
5
= q ˆ
† γ 5 q ˆd
3
x
(12.166)
2
which are pseudoscalars, belonging to the T=1 representation of SU(2). We
have a new non-Abelian global symmetry, called chiral SU(2) f , which we shall
denote by SU(2) f 5 . As far as their action in the isospinor u-d space is concerned, these chiral charges have exactly the same effect as the ordinary flavour
isospin operators of (12.109). But they are pseudoscalars rather than scalars,
and hence they flip the parity of a state on which they act. Thus, whereas
ˆ (
1 )
the isospin raising operator T
2
+
is such that
ˆ (
1 )
T
2
+ |d› = |u›,
(12.167)
ˆ (
1 )
T
2
+ 5 will also produce a u-type state from a d-type one via
ˆ (
1 )
T
2
+ 5 |d› = |u ˜›,
(12.168)
| ›
| ›
ˆ (
1 ) ˆ
but the u ˜ state will have opposite parity from u . Further, since [T
2
+ 5 , H] =
˜
0, this state u ˜ will be degenerate with d . Similarly, the state d produced
ˆ (
1 )
2
| ›
| ›
| ›
via T 5 |u› will have opposite parity from
−
|d›, and will be degenerate with
|u›. The upshot is that we have two massless states u , d of (say) positive
˜
| › | ›
parity, and a further two massless states |u ˜›, |d› of negative parity, in this
simple model.
Suppose we now let the quarks interact, for example by an interaction of
the QCD type, already indicated in (12.136). In that case, the interaction
terms have the form
λ
¯
uγ ˆ
μ a
¯
ˆ
ˆ
ˆA
a
λ
ˆ
a ˆ
u μ + dγ
μ
dA
a
2
2
μ
(12.169)
where
(
)
(
)
ˆ
u ˆ r
d r
u ˆ = ( u ˆ ) ˆ
, d = ( ˆ
b
d )
b
(12.170)
u ˆ g
ˆ
d g
and the 3 × 3 λ’s act in the r-b-g space. Just as in the previous U(1) case,
the interaction (12.169) is invariant under the global SU(2) f 5 chiral symmetry
(12.164), acting in the u-d space. Note that, somewhat confusingly, (12.169) is
not a simple ‘gauging’ of (12.163): a covariant derivative is being introduced,
but in the space of a new (colour) degree of freedom, not in flavour space. In
fact, the flavour degrees of freedom are ‘inert’ in (12.169), so that it is invariant
under SU(2) f transformations, while the Dirac structure implies that it is also
invariant under chiral SU(2) f 5 transformations (12.164). All the foregoing can
be extended unchanged to chiral SU(3) f 5 , given that QCD is ‘flavour blind’,
and supposing that m s ≈ 0.