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-

ˆ

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

ˆ

[φ(x), π (y)] = i δ d’1 (x ’ y).

ˆ

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

completely.

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-

ground.

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,

1

S(φ) = dd x φ(x)(’∇2 + m2 )φ(x) + VI φ(x) , (1)

x

2

Chiral Anomalies and Topology 171

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

1

∆B (p) =

p 2 + m2 reg.

with

n

∆’1 (p) 2 2

(1 + p2 /Mi2 ).

= (p + m ) (2)

B

i=1

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-

ization.

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

∞

2

+m2 )

dt ρ(tΛ2 )e’t(p

∆B (p) = , (3)

0

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

n

∆’1 (p) (1 + p2 /Mi2 ).

= (m + ip) (5)

F

i=1

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

1

Z(J) = det1/2 (∆B ) exp [’VI (δ/δJ)] exp dd x dd y J(x)∆B (x ’ y)J(y) ,

2

(6)

where the determinant is generated by the gaussian integration, and