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ln det eiγ5 θ(x)Deiγ5 θ(x) = ln det D ’ i d4 x θ(x) µνρσ Fµν (x)Fρσ (x). (29)

Remark. One might be surprised that in the calculation the divergence of the
axial current does not vanish, though the regularization of the fermion propaga-
tor seems to be consistent with chiral symmetry. The reason is simple: if we add
for example higher derivative terms to the action, the form of the axial current
is modi¬ed and the additional contributions cancel the term we have found.
In the form we have organized the calculation, it generalizes without di¬-
culty to general even dimensions 2n. Note simply that the permutation (p1 , µ) ”
(p2 , ν) in (25) is replaced by a cyclic permutation. If gauge invariance is main-
tained, the anomaly in the divergence of the axial current J» (x) in general is

= ’2i
‚» J» (x) µ1 ν1 ...µn νn Fµ1 ν1 . . . Fµn νn , (30)
(4π)n n!
where µ1 ν1 ...µn νn is the completely antisymmetric tensor, and J» ≡ J»
the axial current.

Boson Regulator Fields. We have seen that we could also regularize by
adding massive fermions and bosons with fermion spin, the unpaired boson af-
fecting transformation properties under space-dependent chiral transformations.
Denoting by φ the boson ¬eld and by M its mass, we perform in the regularized
192 J. Zinn-Justin

functional integral a change of variables of the form of a space-dependent chi-
ral transformation acting in the same way on the fermion and boson ¬eld. The
variation δS of the action at ¬rst order in θ is

d4 x ‚µ θ(x)Jµ (x) + 2iM θ(x)φ(x)γ5 φ(x)
δS =

¯ ¯
Jµ (x) = iψ(x)γ5 γµ ψ(x) + iφ(x)γ5 γµ φ(x).
Expanding in θ and identifying the coe¬cient of θ(x), we thus obtain the equa-
‚µ Jµ (x) = 2iM φ(x)γ5 φ(x) = ’2iM tr γ5 x| D’1 |x .
The divergence of the axial current comes here from the boson contribution. We
know that in the large M limit it becomes quadratic in A. Expanding the r.h.s. in
powers of A, keeping the quadratic term, we ¬nd after Fourier transformation
d4 q
(k) = ’2iM e
(2) 2 4 4
C d p1 d p2 Aµ (p1 )Aν (p2 )
— tr γ5 (q + k ’ iM )’1 γµ (q ’ p 2 ’ iM )’1 γν (q ’ iM )’1 . (32)

The apparent divergence of this contribution is regularized by formally vanishing
diagrams that we do not write, but which justify the following formal manipu-
In the trace the formal divergences cancel and one obtains

C (2) (k) ∼M ’∞ 8M 2 e2 d4 p1 d4 p2 p1ρ p2σ Aµ (p1 )Aν (p2 )

d4 q
— .
(2π)4 (q 2 + M 2 )3
The limit M ’ ∞ corresponds to remove the regulator. The limit is ¬nite
because after rescaling of q the mass can be eliminated. One ¬nds
(k) ∼
d4 p1 d4 p2 p1ρ p2σ Aµ (p1 )Aν (p2 ) ,
C µνρσ
M ’∞ 4π 2

in agreement with (27).

Point-Splitting Regularization. Another calculation, based on regulariza-
tion by point splitting, gives further insight into the mechanism that generates
the anomaly. We thus consider the non-local operator
= iψ(x ’ a/2)γ5 γµ ψ(x + a/2) exp ie
Jµ (x, a) A» (s)ds» ,

in the limit |a| ’ 0. To avoid a breaking of rotation symmetry by the regulariza-
tion, before taking the limit |a| ’ 0 we will average over all orientations of the
Chiral Anomalies and Topology 193

vector a. The multiplicative gauge factor (parallel transporter) ensures gauge
invariance of the regularized operator (transformations (19)). The divergence of
the operator for |a| ’ 0 then becomes

‚µ Jµ (x, a) ∼ ’ea» ψ(x ’ a/2)γ5 γµ Fµ» (x)ψ(x + a/2)

— exp ie A» (s)ds» ,

where the ψ, ψ ¬eld equations have been used. We now expand the expectation
value of the equation in powers of A. The ¬rst term vanishes. The second term
is quadratic in A and yields

‚µ Jµ (x, a) ∼ ie2 a» Fµ» (x)
d4 y Aν (y+x) tr γ5 ∆F (y’a/2)γν ∆F (’y’a/2)γµ ,

where ∆F (y) is the fermion propagator:

i k 1y
∆F (y) = ’ d4 k eiky = .
(2π)4 k2 2π 2 y 4

We now take the trace. The propagator is singular for |y| = O(|a|) and, therefore,
we can expand Aν (x + y) in powers of y. The ¬rst term vanishes for symmetry
reasons (y ’ ’y), and we obtain

ie2 yρ yσ a„
d4 y
‚µ Jµ (x, a) µν„ σ a» Fµ» (x)‚ρ Aν (x) .
|y + a/2|4 |y ’ a/2|4

The integral over y gives a linear combination of δρσ and aρ aσ but the second
term gives a vanishing contribution due to symbol. It follows that

y 2 ’ (y · a)2 /a2

x5 4
‚µ Jµ (x, a) µν„ ρ a» a„ Fµ» (x)‚ρ Aν (x) dy .
|y + a/2|4 |y ’ a/2|4
3π 4

After integration, we then ¬nd

ie2 a» a„
‚µ Jµ (x, a) ∼
Fµ» (x)Fρν (x).
µν„ ρ
4π 2 a2
Averaging over the a directions, we see that the divergence is ¬nite for |a| ’ 0
and, thus,
lim ‚µ Jµ (x, a) = µν»ρ Fµ» (x)Fρν (x),
16π 2

in agreeement with the result (28).
On the lattice an averaging over aµ is produced by summing over all lattice
directions. Because the only expression quadratic in aµ that has the symmetry
of the lattice is a2 , the same result is found: the anomaly is lattice-independent.
194 J. Zinn-Justin

A Direct Physical Application. In a phenomenological model of Strong
Interaction physics, where a SU (2) — SU (2) chiral symmetry is softly broken by
the pion mass, in the absence of anomalies the divergence of the neutral axial
current is proportional to the π0 ¬eld (corresponding to the neutral pion). A
short formal calculation then indicates that the decay rate of π0 into two photons
should vanish at zero momentum. Instead, taking into account the axial anomaly
(28), one obtains a non-vanishing contribution to the decay, in good agreement
with experimental data.

Chiral Gauge Theory. A gauge theory is consistent only if the gauge ¬eld
is coupled to a conserved current. An anomaly that a¬ects the current destroys
gauge invariance in the full quantum theory. Therefore, the theory with axial
gauge symmetry, where the action in the fermion sector reads

¯ ¯
S(ψ, ψ; B) = ’ d4 x ψ(x)(‚ + igγ5B)ψ(x),

is inconsistent. Indeed current conservation applies to the BBB vertex at one-
loop order. Because now the three point vertex is symmetric the divergence is
given by the expression (26), and thus does not vanish.
More generally, the anomaly prevents the construction of a theory that would
have both an abelian gauge vector and axial symmetry, where the action in the
fermion sector would read

¯ ¯
S(ψ, ψ; A, B) = ’ d4 x ψ(x)(‚ + ieA + iγ5 gB)ψ(x).

A way to solve both problems is to cancel the anomaly by introducing another
fermion of opposite chiral coupling. With more fermions other combinations of
couplings are possible. Note, however, that a purely axial gauge theory with
two fermions of opposite chiral charges can be rewritten as a vector theory by
combining di¬erently the chiral components of both fermions.

4.3 Two Dimensions

As an exercise and as a preliminary to the discussion of the CP (N ’ 1) models
in Sect.5.2, we verify by explicit calculation the general expression (30) in the
special example of dimension 2:
‚µ Jµ = ’i
µν Fµν . (33)

The general form of the r.h.s. is again dictated by locality and power counting:
the anomaly must have canonical dimension 2. The explicit calculation requires
some care because massless ¬elds may lead to IR divergences in two dimensions.
One thus gives a mass m to fermions, which breaks chiral symmetry explicitly,
Chiral Anomalies and Topology 195

and takes the massless limit at the end of the calculation. The calculation involves
only one diagram:

δ δ
i tr γ3 γµD’1 (k)
“µν (k, ’k) =
(2) 3
Jµ (k) =
δAν (’k) δAν (’k)
A=0 A=0
e 1 1
tr γ3 γµ d2 q
= γν .
(2π)2 iq + m iq + ik + m

Here the γ-matrices are simply the ordinary Pauli matrices. Then,

e 1 1
kµ “µν (k, ’k) =
d2 q
tr γ3 k γν .
(2π)2 iq + m iq + ik + m

We use the method of the boson regulator ¬eld, which yields the two-dimensional
analogue of (31). Here, it leads to the calculation of the di¬erence between two
diagrams (analogues of (32)) due to the explicit chiral symmetry breaking:

e 1 1
’ (m ’ M )
d2 q
Cµ (k) = 2m tr γ3 γν
(2π)2 iq + m iq + ik + m
(m ’ iq) γµ (m ’ iq ’ ik)
’ (m ’ M ).
d2 q 2
= 2m tr γ3
(2π)2 (q + m2 )[(k + q)2 + m2 ]

In the trace again the divergent terms cancel:

d2 q
’ (m ’ M ).
Cµ (k) = 4em µν kν


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