Chapter 5
Anomalies
When classical symmetry can not be maintained in QFT, we say we have an anomaly.
Note, this is quite different from spontenous symmetry breaking.
We are just discussed one type, anomaly in scale invariance. Mass without mass
(dimensional transmutation). PQCD unification scale/renormalization.
Another kind, with a rather difficult flavor is connected with chiral symmetry.
Many applications:
• Eliminate extra UA (1) symmetry of massless QCD (approximate for real QCD,
but still too good).
• Eliminate I3S (axial isospin) of QCD × QCD ⇒ mn� >> mπ (actually, modify
it) π ◦ → 2γ
• Constraint on what QFTs are consistent by demanding no anomalies in gauge
symmetries.
• Connections with topology/solitons.
• ···
• Hawking radiation (recent work).
A
Begin with a very concrete low basis approach. Δ V V graph for mass fermions.
I λµν ≡ i
�
d4 p
1
1
1
tr(γ λ γ 5
γν
γ µ + crossed)
4
(2π)
p/ − /q p/ − k/1 p/
(5.1)
?
Check if k1 µI λµν = 0
1
1
1
1
=
−
k/1
/
/
p/ − k 1 p/
p/ − k 1 p/
47
(5.2)
48
CHAPTER 5. ANOMALIES
q
λ
p−q
S
p
+ Crossed (k1,µ
k2, ν )
p−k1
k2,ν
k1,µ
Figure 5.1: Low Basis.
1
1
1
1
(/q − k/2 )
=
(5.3)
−
p/ − /q
p/ − /q p/ − k/2
p/ − k/2
�
d4 p
1
1
1
1
tr[γ λ γ 5 (
γν
k1µ I λµν = i
−
γ ν )] (5.4)
4
(2π)
p/ − /q p/ − k/1 p/ − k/2 p/
�
This is superficially linearly divergent (note pp → 0 in numerator). Formally,
p → p − k1 in 2nd term makes them cancel. But, linear divergence is dangerous –
shifts can leave finite surface terms at |p| → ∞.
Careful shift:
f (p + a) ≈
f (p) + aµ
� �� �
�
cancels
�
d4 p[f (p + a) − f (p)] =
lim
p→∞
�
f (p) = trγ λ γ 5
aµ = k1µ
k1µ I λµν =
=
∂
f (p) + highers
� �� �
∂pµ
�� � �
→∞
f inite
dΩ
����
angular average
(5.5)
pµ
f (p)2π 2 p3
p
µ
4i�
(5.6)
f rom euclidean rotation
τ νσλ
1
1
4i�
k2τ pσ
γν =
p − k2 p
(p − k2 )2 p2
lim dΩ
p→∞
ia
����
(5.7)
(5.8)
τ νσλ
k2τ (pσ pµi ) 2 3 µ
2π p ik1
p4 (2π)2
i λντ σ
�
k1τ k2σ (=
� 0)
8π 2
(5.9)
However the whole calculation is bogous because we could shift by anything in
internal momentum.
49
i σνµλ
�
aσ + crossed
(5.10)
8π 2
e.g., choosing aσ = 12 (k2 − k1 ) gives conceived V , (or CV C). This corresponds to
symmetric integration.
I λµν (p → p + a, k1 , k2 ) − I λµν (k1 , k2 ) =
p+ q
− 2
Figure 5.2: Symmetric Integration.
With this choice, though, the naive axial divergence gets doubled, just cancelled.
q
i
(2π ) 4
k2
k1
Figure 5.3: Axial Divergence.
1
1
1
1
1
1
= (p/ − (p/ − /q))γ5
= γ5
+ γ5
/qγ5
p/
p/ − /q
p/ − /q
p/ − /q p/
p/
As before, but with λ ↔ µ
i � d4 p
1
1
γ
trγ
γµ = k1µ [�µνλσ k2σ ]
5
ν
(2π)4 (2π)4
p − q p − k1
= − (−(k1 + k2 )λ )[�µνλσ k2σ ]
This leads to
�
��
qλ
�
(5.11)
(5.12)
50
CHAPTER 5. ANOMALIES
∂µ J µS =
e2 µνλσ
�
Fµν Fλσ
(4π)2
(5.13)
2 fields, 4 terms ⇒ factor γ.
Not a very satisfactory derivation, of course. Much better is to use Pauli-Villars
regulator, add massive spin 21 boson for opposite sign in loop. Take M → ∞ at the
end, because we do not want this in the physical spectrum.
Now we can shift with a clean conscience
trγ λ γ5
1
1
1
γν
γµ
p/ − /q − M p/ − k/1 − M p/ − M
(5.14)
where k/1 = (p/ − M ) − (p/ − k/1 − M ), vector is no problem.
Axial vector
(p/ − M ) − (p/ − /q − M )
(5.15)
p/ − /
q + M
1
1
1
1
= γ5
+
γ5
/qγ5
p/ − M
p/ − M p/ − /
q − M
p/ − /q − M
p/ − /q − M
1
1
= odd term + 2M
γ5
(5.16)
p/ − M p/ + /q − M
�
1
1
1
γν
γµ
answer ∝ M d4 ptrγ 5
p/ − /q − M p/ − k/1 − M p/ − M
M �µνλσ (p→0 , k1 , k2 )
sim
(f inite as M → ∞)
(5.17)
(p2 − M 2 )3
/q
=
This gives the same final answer, of course.