±

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

4π

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-

S

tained, the anomaly in the divergence of the axial current J» (x) in general is

en

= ’2i

S

‚» J» (x) µ1 ν1 ...µn νn Fµ1 ν1 . . . Fµn νn , (30)

(4π)n n!

(2n+1)

where µ1 ν1 ...µn νn is the completely antisymmetric tensor, and J» ≡ J»

S

is

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)

5

δS =

with

¯ ¯

5

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-

tion

‚µ Jµ (x) = 2iM φ(x)γ5 φ(x) = ’2iM tr γ5 x| D’1 |x .

¯

5

(31)

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 )

(2π)4

— 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-

lations.

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 )

µνρσ

d4 q

1

— .

(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

e2

(k) ∼

(2)

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

x+a/2

¯

= iψ(x ’ a/2)γ5 γµ ψ(x + a/2) exp ie

5

Jµ (x, a) A» (s)ds» ,

x’a/2

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)

x5

x+a/2

— exp ie A» (s)ds» ,

x’a/2

¯

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)

x5

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„

∼4

x5

d4 y

‚µ Jµ (x, a) µν„ σ a» Fµ» (x)‚ρ Aν (x) .

|y + a/2|4 |y ’ a/2|4

π

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

ie2

∼

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) ∼

x5

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,

ie2

x5

lim ‚µ Jµ (x, a) = µν»ρ Fµ» (x)Fρν (x),

16π 2

|a|’0

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:

e

‚µ Jµ = ’i

3

µν Fµν . (33)

2π

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) =

(2)

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)

e

’ (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

1

’ (m ’ M ).

2

Cµ (k) = 4em µν kν