look at three flavors of free massless
fermions coupled to flavor currents
67
a
μ
q
q
q
v
α
v
β
v
μ
= (
‘
μ
+
r
μ
)
/
2
a
μ
= (
‘
μ
-
r
μ
)
/
2
as usual just using the classical sources as tools to calculate
n
-point functions
clasically there is an
SU
(3)
×
SU
(3)
symmetry so we would expect the axial
current to be conserved — but the graph has IR (as well as UV) divergences
calculate straightforwardly but use Lorentz invariance to pull out a piece
proportional to several factors of momentum so the loop integration is finite
(done in Zee’s book)
or get it as a limit of massive fermions
68
∂
μ
a
μ
q
q
q
v
α
v
β
=
mγ
5
q
q
q
v
α
v
β
but without the
m
, the right hand side is our favorite calculation again — it has a
1
/m
that cancels the
m
in the numerator — thus this contribution persists even as
m
→
0
, leading to an “anomalous divergence” of the axial vector current at
m
= 0
— the anomaly is the fact that the effect persists to
m
= 0
— it’s an IR
effect — so do we need some kind of extra term in our theory to account for this?
for free massless fermions, obviously not — the triangle graph is there in the
theory to produce this effect
in our
Φ
theory — still not — we added massless fermions, which produce the
anomalous divergence — we coupled them to
Φ
and the
Φ
VEV generated a
mass — but as we have seen, the mass and the details of the coupling just cancels
in the triangle diagram which still produces the “anomalous” divergence
but if we integrate the fermions out of the theory - or if we do our chiral rotation
so the fermions transform only under the
u SU
(3)
then we need a new term
69
this resolution of the puzzle is called the anomaly — loops of chiral fermions
coupled to gauge fields break chiral symmetries — integrating out chiral
fermions, either massless fermions or fermions whose mass is generated by
spontaneous breaking of the chiral symmetry, introduces a new term in the low
energy
L
that is not invariant under chiral
SU
(3)
×
SU
(3)
symmetry - the chiral
symmetry is broken by quantum effects — a part of this extra term in
L
involves
the photon fields, and under a
T
3
chiral transformation changes by
-
c
3
3
e
2
16
π
2
tr(
T
3
Q
2
)
²
μνλσ
F
μν
F
λσ
+
· · ·
(the factor of 3 comes from the the three colors of the quarks) — but the effects
of chiral symmetry have not completely disappeared — under a GLOBAL
T
3
chiral transformation
π
0
→
π
0
+
c
, the change in this term is a total derivative —
(like the
²
μναβ
F
μν
F
αβ
term) so the action is invariant if we can neglect the fields
at infinity — the anomaly is an IR effect that only breaks the global chiral
symmetry at long distances
various ways of seeing what this extra term is — call it
W
(
U, v
μ
, a
μ
)
— easiest
to find its change under infinitesimal chiral transformation
ξ
=
e
ic
= 1 +
ic
+
· · ·
for general sources
v
μ
and
a
μ
, Wess and Zumino determined its form from the
SU
(3)
×
SU
(3)
symmetry and Bardeen calculated it directly by looking at all
