85
14.2. The dynamics of colour
there is in fact one more gauge invariant term of mass dimension 4 which can
be written down, namely
ˆ
θg s
2
F ˆ μν F ˆ ρσ
L θ =
∈ μνρσ a a ;
(14.40)
64π 2
this is the ‘θ-term’ of QCD. A full discussion of this term (see for example
Weinberg 1996, section 23.6) is beyond our scope, but we shall give a brief
introduction to the main ideas.
The reader may wonder, first of all, whether the θ-term should give rise
to a new Feynman rule. The answer to this begins by noting that (14.40) can
actually be written as a total divergence:
F ˆ μν F ˆ ρσ
K ˆ μ
∈ μνρσ
= ∂ μ
.
(14.41)
a
a
F ˆ μν F ˆ ρσ
This is more easily seen in the analogous term for QED, namely ∈ μνρσ
.
We have
F ˆ μν F ˆ ρσ
A
ν
− ∂
ν ˆ
A
σ
− ∂
σ A ˆ ρ )
∈ μνρσ
= ∈ μνρσ (∂
μ ˆ
A
μ )(∂
ρ ˆ
(14.42)
= 4∈ μνρσ ∂
μ A ˆ ν ∂
ρ A ˆ σ
(14.43)
= ∂
μ (4∈ μνρσ A ˆ ν ∂
ρ A ˆ σ ),
(14.44)
where we have used the antisymmetry of the ∈ symbol in (14.43), and also in
(14.44) since the contraction of ∈ with the symmetric tensor ∂
μ ∂
ρ vanishes.
We shall not need the explicit form of K ˆ μ .
Any total divergence in a Lagrangian can be integrated to give only a
‘surface’ term in the action, which can usually be discarded, making conventional assumptions about the vanishing of the fields at spatial infinity. There
are, however, field configurations (‘instantons’) which do contribute to the
θ-term. Such configurations are not reachable in perturbation theory, and so
no perturbative Feynman rules are associated with (14.40). They approach
a pure gauge form at spatial infinity, and are therefore associated with the
QCD vacuum state; their effect is equivalent to including the term (14.40) in
the QCD Lagrangian (see for example Rajaraman 1982).
The term (14.40) has potentially important phenomenological implications, since it conserves C but violates both P and T (and hence also CP).
Again, this is easy to see in the QED analogue term (14.42), which equals
8E ˆ · B ˆ (problem 14.3): we recall that under P, E ˆ → −E ˆ and B ˆ → B ˆ , while
under T, E ˆ → E ˆ and B ˆ → −B ˆ . But we know (section 4.2) that strong interactions conserve both P and T to a high degree of accuracy. In particular,
the neutron electric dipole moment d n , which would violate both P and T, is
extremely small (see (4.133)). A very crude estimate of the size of d n , induced
by the θ-term, is given by dimensional analysis as
e
d n ∼
θ,
(14.45)
M n
14.2. The dynamics of colour
there is in fact one more gauge invariant term of mass dimension 4 which can
be written down, namely
ˆ
θg s
2
F ˆ μν F ˆ ρσ
L θ =
∈ μνρσ a a ;
(14.40)
64π 2
this is the ‘θ-term’ of QCD. A full discussion of this term (see for example
Weinberg 1996, section 23.6) is beyond our scope, but we shall give a brief
introduction to the main ideas.
The reader may wonder, first of all, whether the θ-term should give rise
to a new Feynman rule. The answer to this begins by noting that (14.40) can
actually be written as a total divergence:
F ˆ μν F ˆ ρσ
K ˆ μ
∈ μνρσ
= ∂ μ
.
(14.41)
a
a
F ˆ μν F ˆ ρσ
This is more easily seen in the analogous term for QED, namely ∈ μνρσ
.
We have
F ˆ μν F ˆ ρσ
A
ν
− ∂
ν ˆ
A
σ
− ∂
σ A ˆ ρ )
∈ μνρσ
= ∈ μνρσ (∂
μ ˆ
A
μ )(∂
ρ ˆ
(14.42)
= 4∈ μνρσ ∂
μ A ˆ ν ∂
ρ A ˆ σ
(14.43)
= ∂
μ (4∈ μνρσ A ˆ ν ∂
ρ A ˆ σ ),
(14.44)
where we have used the antisymmetry of the ∈ symbol in (14.43), and also in
(14.44) since the contraction of ∈ with the symmetric tensor ∂
μ ∂
ρ vanishes.
We shall not need the explicit form of K ˆ μ .
Any total divergence in a Lagrangian can be integrated to give only a
‘surface’ term in the action, which can usually be discarded, making conventional assumptions about the vanishing of the fields at spatial infinity. There
are, however, field configurations (‘instantons’) which do contribute to the
θ-term. Such configurations are not reachable in perturbation theory, and so
no perturbative Feynman rules are associated with (14.40). They approach
a pure gauge form at spatial infinity, and are therefore associated with the
QCD vacuum state; their effect is equivalent to including the term (14.40) in
the QCD Lagrangian (see for example Rajaraman 1982).
The term (14.40) has potentially important phenomenological implications, since it conserves C but violates both P and T (and hence also CP).
Again, this is easy to see in the QED analogue term (14.42), which equals
8E ˆ · B ˆ (problem 14.3): we recall that under P, E ˆ → −E ˆ and B ˆ → B ˆ , while
under T, E ˆ → E ˆ and B ˆ → −B ˆ . But we know (section 4.2) that strong interactions conserve both P and T to a high degree of accuracy. In particular,
the neutron electric dipole moment d n , which would violate both P and T, is
extremely small (see (4.133)). A very crude estimate of the size of d n , induced
by the θ-term, is given by dimensional analysis as
e
d n ∼
θ,
(14.45)
M n