t ≥ 1/Λ2 .

The Gauge Field Propagator. For the free gauge action in a covariant gauge,

ordinary derivatives can be used because in an abelian theory the gauge ¬eld is

neutral. The tensor Fµν is gauge invariant and the action for the scalar combina-

tion ‚µ Aµ is arbitrary. Therefore, the large momentum behaviour of the gauge

¬eld propagator can be arbitrarily improved by the substitution

Fµν Fµν ’ Fµν P (∇2 /Λ2 )Fµν ,

(‚µ Aµ )2 ’ ‚µ Aµ P (∇2 /Λ2 )‚µ Aµ .

2.4 Non-Abelian Gauge Theories

Compared with the abelian case, the new features of the non-abelian gauge

action are the presence of gauge ¬eld self-interactions and ghost terms. For future

purpose we de¬ne our notation. We introduce the covariant derivative, as acting

on a matter ¬eld,

Dµ = ‚µ + Aµ (x) , (9)

where Aµ is an anti-hermitian matrix, and the curvature tensor

Fµν = [Dµ , Dν ] = ‚µ Aν ’ ‚ν Aµ + [Aµ , Aν ]. (10)

178 J. Zinn-Justin

The pure gauge action then is

1

S(Aµ ) = ’ dd x tr Fµν (x)Fµν (x).

4g 2

In the covariant gauge

1

Sgauge (Aµ ) = ’ dd x tr(‚µ Aµ )2 ,

2ξ

the ghost ¬eld action takes the form

Sghost (Aµ , C, C) = ’

¯ ¯

dd x tr C ‚µ (‚µ C + [Aµ , C]) .

The ghost ¬elds thus have a simple δab /p2 propagator and canonical dimension

one in four dimensions.

The problem of regularization in non-abelian gauge theories shares several

features both with the abelian case and with the non-linear σ-model. The regu-

larized gauge action takes the form

Sreg. (Aµ ) = ’ dd x tr Fµν P D2 Λ2 Fµν ,

in which P is a polynomial of arbitrary degree. In the same way, the gauge

function ‚µ Aµ is changed into

‚µ Aµ ’’ Q ‚ 2 Λ2 ‚µ Aµ ,

in which Q is a polynomial of the same degree as P . As a consequence, both the

gauge ¬eld propagator and the ghost propagator can be arbitrarily improved.

However, as in the abelian case, the covariant derivatives generate new inter-

actions that are more singular. It is easy to verify that the power counting of

one-loop diagrams is unchanged while higher order diagrams can be made con-

vergent by taking the degrees of P and Q large enough: Regularization by higher

derivatives takes care of all diagrams except, as in non-linear σ models, some

one-loop diagrams (and thus subdiagrams).

As with charged matter, the one-loop diagrams have to be examined sepa-

rately. However, for fermion matter it is still possible as, in the abelian case, to

add a set of regulator ¬elds, massive fermions and bosons with fermion spin. In

the chiral situation, the problem of the compatibility between the gauge sym-

metry and the quantization is reduced to an explicit veri¬cation of the Ward“

Takahashi (WT) identities for the one-loop diagrams. Note that the preserva-

tion of gauge symmetry is necessary for the cancellation of unphysical states in

physical amplitudes and, thus, essential to ensure the physical relevance of the

quantum ¬eld theory.

3 Other Regularization Schemes

The other regularization schemes we now discuss, have the common property

that they modify in some essential way the structure of space“time: dimensional

Chiral Anomalies and Topology 179

regularization because it relies on de¬ning Feynman diagrams for non-integer

dimensions, lattice regularization because continuum space is replaced by a dis-

crete lattice.

3.1 Dimensional Regularization

Dimensional regularization involves continuation of Feynman diagrams in the pa-

rameter d (d is the space dimension) to arbitrary complex values and, therefore,

seems to have no meaning outside perturbation theory. However, this regular-

ization very often leads to the simplest perturbative calculations.

In addition, it solves the problem of commutation of quantum operators in

local ¬eld theories. Indeed commutators, for example in the case of a scalar ¬eld,

take the form (in the Schr¨dinger picture)

o

ˆ

[φ(x), π (y)] = i δ d’1 (x ’ y) = i (2π)1’d dd’1 p eip(x’y) ,

ˆ

ˆ

where π (x) is the momentum conjugate to the ¬eld φ(x). As we have already

ˆ

stressed, in a local theory all operators are taken at the same point and, therefore,

ˆπ

a commutation in the product φ(x)ˆ (x) generates a divergent contribution (for

d > 1) proportional to

δ d’1 (0) = (2π)1’d dd’1 p .

The rules of dimensional regularization imply the consistency of the change of

variables p ’ »p and thus

dd p ’

dd p = »d dd p = 0 ,

in contrast to momentum regularization where it is proportional to a power of the

cut-o¬. Therefore, the order between operators becomes irrelevant because the

commutator vanishes. Dimensional regularization thus is applicable to geometric

models where these problems of quantization occur, like non-linear σ models or

gauge theories.

Its use, however, requires some care in massless theories. For instance, in a

massless theory in two dimensions, integrals of the form dd k/k 2 are met. They

also vanish in dimensional regularization for the same reason. However, here

they correspond to an unwanted cancellation between UV and IR logarithmic

divergences.

More important here, it is not applicable when some essential property of

the ¬eld theory is speci¬c to the initial dimension. An example is provided by

theories containing fermions in which Parity symmetry is violated.

Fermions. For fermions transforming under the fundamental representation of

the spin group Spin(d), the strategy is the same. The evaluation of diagrams with

180 J. Zinn-Justin

fermions can be reduced to the calculation of traces of γ matrices. Therefore, only

one additional prescription for the trace of the unit matrix is needed. There is no

natural continuation since odd and even dimensions behave di¬erently. Since no

algebraic manipulation depends on the explicit value of the trace, any smooth

continuation in the neighbourhood of the relevant dimension is satisfactory. A

convenient choice is to take the trace constant. In even dimensions, as long as

only γµ matrices are involved, no other problem arises. However, no dimensional

continuation that preserves all properties of γd+1 , which is the product of all

other γ matrices, can be found. This leads to serious di¬culties if γd+1 in the

calculation of Feynman diagrams has to be replaced by its explicit expression in

terms of the other γ matrices. For example, in four dimensions γ5 is related to

the other γ matrices by

4! γ5 = ’ µ1 ...µ4 γµ1 . . . γµ4 , (11)

where µ1 ···µ4 is the complete antisymmetric tensor with 1234 = 1. Therefore,

problems arise in the case of gauge theories with chiral fermions, because the

special properties of γ5 are involved as we recall below. This di¬culty is the

source of chiral anomalies.

Since perturbation theory involves the calculation of traces, one possibility

is to de¬ne γ5 near four dimensions by

γ5 = Eµ1 ...µ4 γµ1 . . . γµ4 , (12)

where Eµνρσ is a completely antisymmetric tensor, which reduces to ’ µνρσ /4!

in four dimensions. It is easy to then verify that, with this de¬nition, γ5 anticom-

mutes with the other γµ matrices only in four dimensions. If, for example, one

evaluates the product γν γ5 γν in d dimensions, replacing γ5 by (12) and using

systematically the anticommutation relations γµ γν + γν γµ = 2δµν , one ¬nds

γν γ5 γν = (d ’ 8)γ5 .

Anticommuting properties of the γ5 would have led to a factor ’d, instead.

This additional contribution, proportional to d ’ 4, if it is multiplied by a factor

1/(d ’ 4) consequence of UV divergences in one-loop diagrams, will lead to a

¬nite di¬erence with the formal result.

The other option would be to keep the anticommuting property of γ5 but the

preceding example shows that this is contradictory with a form (12). Actually,

one veri¬es that the only consistent prescription for generic dimensions then is

that the traces of γ5 with any product of γµ matrices vanishes and, thus, this

prescription is useless.

Finally, an alternative possibility consists in breaking O(d) symmetry and

keeping the four γ matrices of d = 4.

3.2 Lattice Regularization

We have explained that Pauli“Villars™s regularization does not provide a com-

plete regularization for ¬eld theories in which the geometric properties generate

Chiral Anomalies and Topology 181

interactions like models where ¬elds belong to homogeneous spaces (e.g. the non-

linear σ-model) or non-abelian gauge theories. In these theories some divergences

are related to the problem of quantization and order of operators, which already

appears in simple quantum mechanics. Other regularization methods are then

needed. In many cases lattice regularization may be used.

Lattice Field Theory. To each site x of a lattice are attached ¬eld variables

corresponding to ¬elds in the continuum. To the action S in the continuum

corresponds a lattice action, the energy of lattice ¬eld con¬gurations in the lan-

guage of classical statistical physics. The functional integral becomes a sum over

con¬gurations and the regularized partition function is the partition function of

a lattice model.

All expressions in these notes will refer implicitly to a hypercubic lattice and

we denote the lattice spacing by a.

The advantages of lattice regularization are:

(i) Lattice regularization indeed corresponds to a speci¬c choice of quantiza-

tion.

(ii) It is the only established regularization that for gauge theories and other

geometric models has a meaning outside perturbation theory. For instance the

regularized functional integral can be calculated by numerical methods, like

stochastic methods (Monte-Carlo type simulations) or strong coupling expan-

sions.

(iii) It preserves most global and local symmetries with the exception of the

space O(d) symmetry, which is replaced by a hypercubic symmetry (but this

turns out not to be a major di¬culty), and fermion chirality, which turns out to

be a more serious problem, as we will show.

The main disadvantage is that it leads to rather complicated perturbative

calculations.

3.3 Boson Field Theories

Scalar Fields. To the action (1) for a scalar ¬eld φ in the continuum corre-

sponds a lattice action, which is obtained in the following way: The euclidean

lagrangian density becomes a function of lattice variables φ(x), where x now is a

lattice site. Locality can be implemented by considering lattice lagrangians that

depend only on a site and its neighbours (though this is a too strong require-

ment; lattice interactions decreasing exponentially with distance are also local).

Derivatives ‚µ φ of the continuum are replaced by ¬nite di¬erences, for example:

‚µ φ ’ ∇lat. φ = [φ(x + anµ ) ’ φ(x)] /a , (13)

µ

where a is the lattice spacing and nµ the unit vector in the µ direction. The

lattice action then is the sum over lattice sites.

182 J. Zinn-Justin

With the choice (13), the propagator ∆a (p) for the Fourier components of a

massive scalar ¬eld is given by

d

2

∆’1 (p) 1 ’ cos(apµ ) .

2

=m + 2

a

a µ=1

It is a periodic function of the components pµ of the momentum vector with

period 2π/a. In the small lattice spacing limit, the continuum propagator is

recovered:

∆’1 (p) = m2 + p2 ’ 12 a2 p4 + O p6 .

1

a µ µ

µ

In particular, hypercubic symmetry implies O(d) symmetry at order p2 .

Gauge Theories. Lattice regularization de¬nes unambiguously a quantum the-

ory. Therefore, once one has realized that gauge ¬elds should be replaced by link

variables corresponding to parallel transport along links of the lattice, one can

regularize a gauge theory.

The link variables Uxy are group elements associated with the links joining

2

the sites x and y on the lattice. The regularized form of dx Fµν is a sum of

products of link variables along closed curves on the lattice. On a hypercubic

lattice, the smallest curve is a square leading to the well-known plaquette action

(each square forming a plaquette). The typical gauge invariant lattice action

corresponding to the continuum action of a gauge ¬eld coupled to scalar bosons

then has the form

S(U, φ— , φ) = β φ— Uxy φy + V (φ— φx ),

tr Uxy Uyz Uzt Utx + κ x x

all all all

plaquettes links sites

(14)

where x, y,... denotes lattice sites, and β and κ are coupling constants. The

action (14) is invariant under independent group transformations on each lattice

site, lattice equivalents of gauge transformations in the continuum theory. The

measure of integration over the gauge variables is the group invariant measure

on each site. Note that on the lattice and in a ¬nite volume, the gauge invariant

action leads to a well-de¬ned partition function because the gauge group (¬nite

product of compact groups) is compact. However, in the continuum or in¬nite

volume limits the compact character of the group is lost. Even on the lattice,

regularized perturbation theory is de¬ned only after gauge ¬xing.

Finally, we note that, on the lattice, the di¬culties with the regularization do

not come from the gauge ¬eld directly but involve the gauge ¬eld only through