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the integration over chiral fermions.

3.4 Fermions and the Doubling Problem
We now review a few problems speci¬c to relativistic fermions on the lattice. We
consider the free action for a Dirac fermion
¯ ¯
S(ψ, ψ) = dd x ψ(x) (‚ + m) ψ(x).
Chiral Anomalies and Topology 183

A lattice regularization of the derivative ‚µ ψ(x), which preserves chiral proper-
ties in the massless limit, is, for example, the symmetric combination

∇lat. ψ(x) = [ψ(x + anµ ) ’ ψ(x ’ anµ )] /2a .

In the boson case, there is no equivalent constraint and thus a possible choice is
the expression 13.
The lattice Dirac operator for the Fourier components ψ(p) of the ¬eld (in-
verse of the fermion propagator ∆lat. (p)) is
sin apµ
Dlat. (p) = m + i γµ , (15)

a periodic function of the components pµ of the momentum vector. A problem
then arises: the equations relevant to the small lattice spacing limit,

sin(a pµ ) = 0 ,

have each two solutions pµ = 0 and pµ = π/a within one period, that is 2d
solutions within the Brillouin zone. Therefore, the propagator (15) propagates
2d fermions. To remove this degeneracy, it is possible to add to the regularized
action an additional scalar term δS involving second derivatives:
¯ ¯ ¯ ¯
2ψ(x)ψ(x) ’ ψ (x + anµ ) ψ(x) ’ ψ(x)ψ (x + anµ ) . (16)
δS(ψ, ψ) = 1 M

The modi¬ed Dirac operator for the Fourier components of the ¬eld reads
(1 ’ cos apµ ) +
DW (p) = m + M γµ sin apµ . (17)
µ µ

The fermion propagator becomes
† †
∆(p) = DW (p) DW (p)DW (p)


(1 ’ cos apµ ) sin2 apµ .
DW (p)DW (p) = m + M +
µ µ

Therefore, the degeneracy between the di¬erent states has been lifted. For each
component pµ that takes the value π/a the mass is increased by 2M . If M is of
order 1/a the spurious states are eliminated in the continuum limit. This is the
recipe of Wilson™s fermions.
However, a problem arises if one wants to construct a theory with massless
fermions and chiral symmetry. Chiral symmetry implies that the Dirac operator
D(p) anticommutes with γ5 :

{D(p), γ5 } = 0 ,
184 J. Zinn-Justin

and, therefore, both the mass term and the term (16) are excluded. It remains
possible to add various counter-terms and try to adjust them to recover chiral
symmetry in the continuum limit. But there is no a priori guarantee that this is
indeed possible and, moreover, calculations are plagued by ¬ne tuning problems
and cancellations of unnecessary UV divergences.
One could also think about modifying the fermion propagator by adding
terms connecting fermions separated by more than one lattice spacing. But it
has been proven that this does not solve the doubling problem. (Formal solutions
can be exhibited but they violate locality that implies that D(p) should be a
smooth periodic function.) In fact, this doubling of the number of fermion degrees
of freedom is directly related to the problem of anomalies.
Since the most naive form of the propagator yields 2d fermion states, one
tries in practical calculations to reduce this number to a smaller multiple of
two, using for instance the idea of staggered fermions introduced by Kogut and
However, the general picture has recently changed with the discovery of the
properties of overlap fermions and solutions of the Ginsparg“Wilson relation or
domain wall fermions, a topic we postpone and we will study in Sect. 7.

4 The Abelian Anomaly
We have pointed out that none of the standard regularization methods can deal
in a straightforward way with one-loop diagrams in the case of gauge ¬elds
coupled to chiral fermions. We now show that indeed chiral symmetric gauge
theories, involving gauge ¬elds coupled to massless fermions, can be found where
the axial current is not conserved. The divergence of the axial current in a chiral
quantum ¬eld theory, when it does not vanish, is called an anomaly. Anomalies
in particular lead to obstructions to the construction of gauge theories when the
gauge ¬eld couples di¬erently to the two fermion chiral components.
Several examples are physically important like the theory of weak electro-
magnetic interactions, the electromagnetic decay of the π0 meson, or the U (1)
problem. We ¬rst discuss the abelian axial current, in four dimensions (the gen-
eralization to all even dimensions then is straightforward), and then the general
non-abelian situation.

4.1 Abelian Axial Current and Abelian Vector Gauge Fields
The only possible source of anomalies are one-loop fermion diagrams in gauge
theories when chiral properties are involved. This reduces the problem to the dis-
cussion of fermions in background gauge ¬elds or, equivalently, to the properties
of the determinant of the gauge covariant Dirac operator.
We thus consider a QED-like fermion action for massless Dirac fermions ψ, ψ
in the background of an abelian gauge ¬eld Aµ of the form

¯ ¯
S(ψ, ψ; A) = ’ D ≡ ‚ + ieA(x) ,
d4 x ψ(x)Dψ(x), (18)
Chiral Anomalies and Topology 185

and the corresponding functional integral

¯ ¯
Z(Aµ ) = dψdψ exp ’S(ψ, ψ; A) = det D .

We can ¬nd regularizations that preserve gauge invariance, that is invariance
under the transformations
ψ(x) = e’iΛ(x) ψ (x),
¯ ¯ Aµ (x) = ’ ‚ν Λ(x) + Aµ (x),
ψ(x) = eiΛ(x) ψ (x),
and, since the fermions are massless, chiral symmetry. Therefore, we would
naively expect the corresponding axial current to be conserved (symmetries are
generally related to current conservation). However, the proof of current con-
servation involves space-dependent chiral transformations and, therefore, steps
that cannot be regularized without breaking local chiral symmetry.
Under the space-dependent chiral transformation
¯ ¯
ψθ (x) = eiθ(x)γ5 ψ(x), ψθ (x) = ψ(x)eiθ(x)γ5 , (20)

the action becomes

¯ ¯ ¯
Sθ (ψ, ψ; A) = ’ d4 x ψθ (x)Dψθ (x) = S(ψ, ψ; A) + d4 x ‚µ θ(x)Jµ (x),

where Jµ (x), the coe¬cient of ‚µ θ, is the axial current:

Jµ (x) = iψ(x)γ5 γµ ψ(x).

After the transformation 20, Z(Aµ ) becomes

Z(Aµ , θ) = det eiγ5 θ(x)Deiγ5 θ(x) .

Note that ln[Z(Aµ , θ)] is the generating functional of connected ‚µ Jµ correlation
functions in an external ¬eld Aµ .
Since eiγ5 θ has a determinant that is unity, one would naively conclude that
Z(Aµ , θ) = Z(Aµ ) and, therefore, that the current Jµ (x) is conserved. This is a

conclusion we now check by an explicit calculation of the expectation value of
‚µ Jµ (x) in the case of the action 18.

(i) For any regularization that is consistent with the hermiticity of γ5

|Z(Aµ , θ)| = det eiγ5 θ(x)Deiγ5 θ(x) det e’iγ5 θ(x)D† e’iγ5 θ(x) = det (DD† ),

and thus |Z(Aµ , θ)| is independent of θ. Therefore, an anomaly can appear only
in the imaginary part of ln Z.
186 J. Zinn-Justin

(ii) We have shown that one can ¬nd a regularization with regulator ¬elds
such that gauge invariance is maintained, and the determinant is independent
of θ for θ(x) constant.
(iii) If the regularization is gauge invariant, Z(Aµ , θ) is also gauge invariant.
Therefore, a possible anomaly will also be gauge invariant.
(iv) ln Z(Aµ , θ) receives only connected, 1PI contributions. Short distance
singularities coming from one-loop diagrams thus take the form of local poly-
nomials in the ¬elds and sources. Since a possible anomaly is a short distance
e¬ect (equivalently a large momentum e¬ect), it must also take the form of a
local polynomial of Aµ and ‚µ θ constrained by parity and power counting. The
¬eld Aµ and ‚µ θ have dimension 1 and no mass parameter is available. Thus,

ln Z(Aµ , θ) ’ ln Z(Aµ , 0) = i d4 x L(A, ‚θ; x),

where L is the sum of monomials of dimension 4. At order θ only one is available:

L(A, ‚θ; x) ∝ e2 µνρσ ‚µ θ(x)Aν (x)‚ρ Aσ (x),

where µνρσ is the complete antisymmetric tensor with = 1. A simple
integration by parts and anti-symmetrization shows that

d4 x L(A, ‚θ; x) ∝ e2 d4 x Fµν (x)Fρσ (x)θ(x),

where Fµν = ‚µ Aν ’ ‚ν Aµ is the electromagnetic tensor, an expression that is
gauge invariant.
The coe¬cient of θ(x) is the expectation value in an external gauge ¬eld of
‚µ Jµ (x), the divergence of the axial current. It is determined up to a multiplica-
tive constant:

‚» J» (x) ∝ e2 ∝ e2
µνρσ ‚µ Aν (x)‚ρ Aσ (x) µνρσ Fµν (x)Fρσ (x) ,

where we denote by • expectation values with respect to the measure e’S(ψ,ψ;A) .
Since the possible anomaly is independent up to a multiplicative factor of
the regularization, it must indeed be a gauge invariant local function of Aµ .
To ¬nd the multiplicative factor, which is the only regularization dependent
feature, it is su¬cient to calculate the coe¬cient of the term quadratic in A in
the expansion of ‚» J» (x) in powers of A. We de¬ne the three-point function
in momentum representation by

δ δ
(3) 5
“»µν (k; p1 , p2 ) = J» (k) , (21)
δAµ (p1 ) δAν (p2 ) A=0
δ δ
i tr γ5 γ»D’1 (k)
= .
δAµ (p1 ) δAν (p2 ) A=0

“ (3) is the sum of the two Feynman diagrams of Fig. 2.
Chiral Anomalies and Topology 187

p1 , µ p1 , µ

k, » k, »
q q

p2 , µ p2 , µ

(a) (b)

Fig. 2. Anomalous diagrams.

The contribution of diagram (a) is:

e2 ’1 ’1
γν q ’1 ,
(a) ’ γµ (q ’ p 2 )
d4 q γ5 γ» (q + k)
tr (22)

and the contribution of diagram (b) is obtained by exchanging p1 , γµ ” p2 , γν .
Power counting tells us that the function “ (3) may have a linear divergence
that, due to the presence of the γ5 factor, must be proportional to »µνρ , sym-


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