84
14. QCD I: Introduction, Tree Graph Predictions, and Jets
in (14.33) are the gauge-fixing and ghost terms, respectively, appropriate to
a gauge field propagator of the form (13.69) (with δ ij replaced by δ ab here).
The Feynman rules following from (14.33) are given in appendix Q.
As remarked in section 12.3.2, the fact that the QCD interactions (14.33)
are ‘flavour-blind’ implies that the global flavour symmetries discussed in
chapter 12 are all preserved by QCD. These include the conservation of each
quark flavour (for example, the number of strange quarks minus the number
of strange antiquarks is conserved); and the symmetries SU(2) f and SU(3) f ,
and the chiral symmetries SU(2) 5f and SU(3) 5f , to the extent that these latter
are good symmetries. Further, (14.33) conserves the discrete symmetries P,
C and T, in a manner quite analogous to QED, already covered in section 7.5.
In the case of P and T, the gluon fields A ˆ aμ have the same transformation
properties as the photon field A ˆ μ , and the (normally ordered) SU(3) c currents
μ
ˆ j = ¯ q ˆ f γ
μ 1
ˆ transform in the same way as the electromagnetic current
fa
2 λ a q f
¯ ˆ
qγ
μ q ˆ, ensuring P and T invariance. Under C, the quark fields transform as
usual according to (7.151). Charge conjugation for the gluon field needs a
little more care. The required rule is
ˆ ˆ C ˆ −1 −λ
∗ ˆ
Cλ a A aμ
=
a A aμ .
(14.36)
The overall minus sign in (14.36) is analogous to that for the photon field
(cf (7.152)). To understand the complex conjugate on the right-hand side of
1
(14.36), recall from (7.153) that the complex scalar field φ ˆ = √ (φ ˆ 1 − iφ ˆ 2 )
2
transforms according to
C ˆ (φ ˆ 1 − iφ ˆ 2 )C ˆ −1 = φ ˆ 1 + iφ ˆ 2 .
(14.37)
μ
Problem 14.2(a) verifies that the (normally ordered) interaction ˆ j A ˆ aμ is then
fa
F ˆ μν
C-invariant. As regards the term F ˆ aμν a , we can write it as
1
μν
Tr(λ a F ˆ aμν λ b F ˆ
b )
(14.38)
2
using the relation
Tr(λ a λ b ) = 2δ ab .
(14.39)
ˆ
A short calculation (problem 14.2(b)) shows that λ a F aμν transforms under
ˆ
C the same way as λ a A aμ (i.e. according to (14.36)). Using the complex
conjugate of (14.39), it then follows that (14.38) is invariant under C.
14.2.4 The θ-term
In arriving at (14.33) we have relied essentially on the ‘gauge principle’ (invariance under a local symmetry) and the requirement of renormalizability (to
forbid the presence of terms with mass dimension higher than 4). The renormalizability of such a theory was proved by ’t Hooft (1971a, b). However,
14. QCD I: Introduction, Tree Graph Predictions, and Jets
in (14.33) are the gauge-fixing and ghost terms, respectively, appropriate to
a gauge field propagator of the form (13.69) (with δ ij replaced by δ ab here).
The Feynman rules following from (14.33) are given in appendix Q.
As remarked in section 12.3.2, the fact that the QCD interactions (14.33)
are ‘flavour-blind’ implies that the global flavour symmetries discussed in
chapter 12 are all preserved by QCD. These include the conservation of each
quark flavour (for example, the number of strange quarks minus the number
of strange antiquarks is conserved); and the symmetries SU(2) f and SU(3) f ,
and the chiral symmetries SU(2) 5f and SU(3) 5f , to the extent that these latter
are good symmetries. Further, (14.33) conserves the discrete symmetries P,
C and T, in a manner quite analogous to QED, already covered in section 7.5.
In the case of P and T, the gluon fields A ˆ aμ have the same transformation
properties as the photon field A ˆ μ , and the (normally ordered) SU(3) c currents
μ
ˆ j = ¯ q ˆ f γ
μ 1
ˆ transform in the same way as the electromagnetic current
fa
2 λ a q f
¯ ˆ
qγ
μ q ˆ, ensuring P and T invariance. Under C, the quark fields transform as
usual according to (7.151). Charge conjugation for the gluon field needs a
little more care. The required rule is
ˆ ˆ C ˆ −1 −λ
∗ ˆ
Cλ a A aμ
=
a A aμ .
(14.36)
The overall minus sign in (14.36) is analogous to that for the photon field
(cf (7.152)). To understand the complex conjugate on the right-hand side of
1
(14.36), recall from (7.153) that the complex scalar field φ ˆ = √ (φ ˆ 1 − iφ ˆ 2 )
2
transforms according to
C ˆ (φ ˆ 1 − iφ ˆ 2 )C ˆ −1 = φ ˆ 1 + iφ ˆ 2 .
(14.37)
μ
Problem 14.2(a) verifies that the (normally ordered) interaction ˆ j A ˆ aμ is then
fa
F ˆ μν
C-invariant. As regards the term F ˆ aμν a , we can write it as
1
μν
Tr(λ a F ˆ aμν λ b F ˆ
b )
(14.38)
2
using the relation
Tr(λ a λ b ) = 2δ ab .
(14.39)
ˆ
A short calculation (problem 14.2(b)) shows that λ a F aμν transforms under
ˆ
C the same way as λ a A aμ (i.e. according to (14.36)). Using the complex
conjugate of (14.39), it then follows that (14.38) is invariant under C.
14.2.4 The θ-term
In arriving at (14.33) we have relied essentially on the ‘gauge principle’ (invariance under a local symmetry) and the requirement of renormalizability (to
forbid the presence of terms with mass dimension higher than 4). The renormalizability of such a theory was proved by ’t Hooft (1971a, b). However,