. 37
( 78 .)


form of anomalies in simple examples. We relate anomalies to the index of the Dirac
operator in a gauge background. We exhibit gauge instantons that contribute to the
anomaly in the example of the CP (N ’1) models and SU (2) gauge theories. We brie¬‚y
mention a few physical consequences. For many years the problem of anomalies had
been discussed only within the framework of perturbation theory. New non-perturbative
solutions based on lattice regularization have recently been proposed. We describe the
so-called overlap and domain wall fermion formulations.

1 Symmetries, Regularization, Anomalies

Divergences. Symmetries of the classical lagrangian or tree approximation do
not always translate into symmetries of the corresponding complete quantum
theory. Indeed, local quantum ¬eld theories are a¬ected by UV divergences that
invalidate simple algebraic proofs.
The origin of UV divergences in ¬eld theory is double. First, a ¬eld contains
an in¬nite number of degrees of freedom. The corresponding divergences are
directly related to the renormalization group and re¬‚ect the property that, even
in renormalizable quantum ¬eld theories, degrees of freedom remain coupled on
all scales.
However, another of type of divergences can appear, which is related to the
order between quantum operators and the transition between classical and quan-
tum hamiltonians. Such divergences are already present in ordinary quantum me-
chanics in perturbation theory, for instance, in the quantization of the geodesic

J. Zinn-Justin, Chiral Anomalies and Topology, Lect. Notes Phys. 659, 167“236 (2005)
http://www.springerlink.com/ c Springer-Verlag Berlin Heidelberg 2005
168 J. Zinn-Justin

motion of a particle on a manifold (like a sphere). Even in the case of forces
linear in the velocities (like a coupling to a magnetic ¬eld), ¬nite ambiguities are
found. In local quantum ¬eld theories the problem is even more severe. For ex-
ample, the commutator of a scalar ¬eld operator φ and its conjugate momentum
π , in the Schr¨dinger picture (in d space“time dimension), takes the form
ˆ o
[φ(x), π (y)] = i δ d’1 (x ’ y).

Hamiltonians contain products of ¬elds and conjugate momenta as soon as
derivative couplings are involved (in covariant theories), or when fermions are
present. Because in a local theory all operators are taken at the same point, prod-
ucts of this nature lead to divergences, except in quantum mechanics (d = 1 with
our conventions). These divergences re¬‚ect the property that the knowledge of
the classical theory is not su¬cient, in general, to determine the quantized theory

Regularization. Regularization is a useful intermediate step in the renormal-
ization program that consists in modifying the initial theory at short distance,
large momentum or otherwise to render perturbation theory ¬nite. Note that
from the point of view of Particle Physics, all these modi¬cations a¬ect in some
essential way the physical properties of the theory and, thus, can only be con-
sidered as intermediate steps in the removal of divergences.
When a regularization can be found which preserves the symmetry of the
initial classical action, a symmetric quantum ¬eld theory can be constructed.
Momentum cut-o¬ regularization schemes, based on modifying propagators
at large momenta, are speci¬cally designed to cut the in¬nite number of degrees
of freedom. With some care, these methods will preserve formal symmetries
of the un-renormalized theory that correspond to global (space-independent)
linear group transformations. Problems may, however, arise when the symmetries
correspond to non-linear or local transformations, like in the examples of non-
linear σ models or gauge theories, due to the unavoidable presence of derivative
couplings. It is easy to verify that in this case regularizations that only cut
momenta do not in general provide a complete regularization.
The addition of regulator ¬elds has, in general, the same e¬ect as modifying
propagators but o¬ers a few new possibilities, in particular, when regulator ¬elds
have the wrong spin“statistics connection. Fermion loops in a gauge background
can be regularized by such a method.
Other methods have to be explored. In many examples dimensional regu-
larization solves the problem because then the commutator between ¬eld and
conjugated momentum taken at the same point vanishes. However, in the case
of chiral fermions dimensional regularization fails because chiral symmetries are
speci¬c to even space“time dimensions.
Of particular interest is the method of lattice regularization, because it can
be used, beyond perturbation theory, either to discuss the existence of a quan-
tum ¬eld theory, or to determine physical properties of ¬eld theories by non-
perturbative numerical techniques. One veri¬es that such a regularization indeed
Chiral Anomalies and Topology 169

speci¬es an order between quantum operators. Therefore, it solves the ordering
problem in non-linear σ-models or non-abelian gauge theories. However, again it
fails in the presence of chiral fermions: the manifestation of this di¬culty takes
the form of a doubling of the fermion degrees of freedom. Until recently, this had
prevented a straightforward numerical study of chiral theories.

Anomalies. That no conventional regularization scheme can be found in the
case of gauge theories with chiral fermions is not surprising since we know the-
ories with anomalies, that is theories in which a local symmetry of the tree or
classical approximation cannot be implemented in the full quantum theory. This
may create obstructions to the construction of chiral gauge theories because
exact gauge symmetry, and thus the absence of anomalies, is essential for the
physical consistency of a gauge theory.
Note that we study in these lectures only local anomalies, which can be deter-
mined by perturbative calculations; peculiar global non-perturbative anomalies
have also been exhibited.
The anomalies discussed in these lectures are local quantities because they are
consequences of short distance singularities. They are responses to local (space-
dependent) group transformations but vanish for a class of space-independent
transformations. This gives them a topological character that is further con-
¬rmed by their relations with the index of the Dirac operator in a gauge back-
The recently discovered solutions of the Ginsparg“Wilson relation and the
methods of overlap and domain wall fermions seem to provide an unconventional
solution to the problem of lattice regularization in gauge theories involving chiral
fermions. They evade the fermion doubling problem because chiral transforma-
tions are no longer strictly local on the lattice (though remain local from the
point of view of the continuum limit), and relate the problem of anomalies with
the invariance of the fermion measure. The absence of anomalies can then be
veri¬ed directly on the lattice, and this seems to con¬rm that the theories that
had been discovered anomaly-free in perturbation theory are also anomaly-free
in the non-perturbative lattice construction. Therefore, the speci¬c problem of
lattice fermions was in essence technical rather than re¬‚ecting an inconsistency
of chiral gauge theories beyond perturbation theory, as one may have feared.
Finally, since these new regularization schemes have a natural implementa-
tion in ¬ve dimensions in the form of domain wall fermions, this again opens the
door to speculations about additional space dimensions.
We ¬rst discuss the advantages and shortcomings, from the point of view
of symmetries, of three regularization schemes, momentum cut-o¬, dimensional,
lattice regularizations. We show that they leave room for possible anomalies
when both gauge ¬elds and chiral fermions are present.
We then recall the origin and the form of anomalies, beginning with the
simplest example of the so-called abelian anomaly, that is the anomaly in the
conservation of the abelian axial current in gauge theories. We relate anomalies
to the index of a covariant Dirac operator in the background of a gauge ¬eld.
170 J. Zinn-Justin

In the two-dimensional CP (N ’ 1) models and in four-dimensional non-
abelian gauge theories, we exhibit gauge instantons. We show that they can be
classi¬ed in terms of a topological charge, the space integral of the chiral anomaly.
The existence of gauge ¬eld con¬gurations that contribute to the anomaly has
direct physical implications, like possible strong CP violation and the solution
to the U (1) problem.
We examine the form of the anomaly for a general axial current, and infer
conditions for gauge theories that couple di¬erently to fermion chiral components
to be anomaly-free. A few physical applications are also brie¬‚y mentioned.
Finally, the formalism of overlap fermions on the lattice and the role of the
Ginsparg“Wilson relation are explained. The alternative construction of domain
wall fermions is explained, starting from the basic mechanism of zero-modes in
supersymmetric quantum mechanics.

Conventions. Throughout these notes we work in euclidean space (with imagi-
nary or euclidean time), and this also implies a formalism of euclidean fermions.
For details see
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon
Press (Oxford 1989, fourth ed. 2002).

2 Momentum Cut-O¬ Regularization
We ¬rst discuss methods that work in the continuum (compared to lattice meth-
ods) and at ¬xed dimension (unlike dimensional regularization). The idea is to
modify ¬eld propagators beyond a large momentum cut-o¬ to render all Feyn-
man diagrams convergent. The regularization must satisfy one important condi-
tion: the inverse of the regularized propagator must remain a smooth function
of the momentum p. Indeed, singularities in momentum variables generate, af-
ter Fourier transformation, contributions to the large distance behaviour of the
propagator, and regularization should modify the theory only at short distance.
Note, however, that such modi¬cations result in unphysical properties of the
quantum ¬eld theory at cut-o¬ scale. They can be considered as intermediate
steps in the renormalization program (physical properties would be recovered
in the large cut-o¬ limit). Alternatively, in modern thinking, the necessity of
a regularization often indicates that quantum ¬eld theories cannot rendered
consistent on all distance scales, and have eventually to be embedded in a more
complete non ¬eld theory framework.

2.1 Matter Fields: Propagator Modi¬cation
Scalar Fields. A simple modi¬cation of the propagator improves the conver-
gence of Feynman diagrams at large momentum. For example in the case of the
action of the self-coupled scalar ¬eld,
S(φ) = dd x φ(x)(’∇2 + m2 )φ(x) + VI φ(x) , (1)
Chiral Anomalies and Topology 171

the propagator in Fourier space 1/(m2 + p2 ) can be replaced by

∆B (p) =
p 2 + m2 reg.

∆’1 (p) 2 2
(1 + p2 /Mi2 ).
= (p + m ) (2)

The masses Mi are proportional to the momentum cut-o¬ Λ,

Mi = ±i Λ , ±i > 0 .

If the degree n is chosen large enough, all diagrams become convergent. In the
formal large cut-o¬ limit Λ ’ ∞, at parameters ± ¬xed, the initial propagator
is recovered. This is the spirit of momentum cut-o¬ or Pauli“Villars™s regular-
Note that such a propagator cannot be derived from a hermitian hamiltonian.
Indeed, hermiticity of the hamiltonian implies that if the propagator is, as above,
a rational function, it must be a sum of poles in p2 with positive residues (as a
sum over intermediate states of the two-point function shows) and thus cannot
decrease faster than 1/p2 .
While this modi¬cation can be implemented also in Minkowski space because
the regularized propagators decrease in all complex p2 directions (except real
negative), in euclidean time more general modi¬cations are possible. Schwinger™s
proper time representation suggests

+m2 )
dt ρ(tΛ2 )e’t(p
∆B (p) = , (3)

in which the function ρ(t) is positive (to ensure that ∆B (p) does not vanish and
thus is invertible) and satis¬es the condition

|1 ’ ρ(t)| < Ce’σt (σ > 0) for t ’ +∞ .

By choosing a function ρ(t) that decreases fast enough for t ’ 0, the behaviour
of the propagator can be arbitrarily improved. If ρ(t) = O(tn ), the behaviour
(2) is recovered. Another example is

ρ(t) = θ(t ’ 1),

θ(t) being the step function, which leads to an exponential decrease:
2 2 2
e’(p +m )/Λ
∆B (p) = . (4)
p 2 + m2
As the example shows, it is thus possible to ¬nd in this more general class prop-
agators without unphysical singularities, but they do not follow from a hamilto-
nian formalism because continuation to real time becomes impossible.
172 J. Zinn-Justin

Spin 1/2 Fermions. For spin 1/2 fermions similar methods are applicable. To
the free Dirac action,

SF0 = dd x ψ(x)(‚ + m)ψ(x) ,

corresponds in Fourier representation the propagator 1/(m + ip). It can be re-
placed by the regularized propagator ∆F (p) where
∆’1 (p) (1 + p2 /Mi2 ).
= (m + ip) (5)

Note that we use the standard notation p ≡ pµ γµ , with euclidean fermion
conventions, analytic continuation to imaginary or euclidean time of the usual
Minkowski fermions, and hermitian matrices γµ .

Remarks. Momentum cut-o¬ regularizations have several advantages: one can
work at ¬xed dimension and in the continuum. However, two potential weak-
nesses have to be stressed:
(i) The generating functional of correlation functions, obtained by adding to
the action (1) a source term for the ¬elds:

S(φ) ’ S(φ) ’ dd x J(x)φ(x),

can be written as

Z(J) = det1/2 (∆B ) exp [’VI (δ/δJ)] exp dd x dd y J(x)∆B (x ’ y)J(y) ,
where the determinant is generated by the gaussian integration, and


. 37
( 78 .)