Σp of the sphere Sp’1 in the same notation can be written as

2π p/2 1

uµ1 duµ2 § . . . § duµp ,

Σp = = µ1 ...µp

(p ’ 1)!

“ (p/2)

when the vector uµ describes the sphere Sp’1 only once. In the r.h.s. of (62), one

thus recognizes an expression proportional to the area of the sphere S3 . Because

in general uµ describes S3 n times when xµ describes S3 only once, a factor n is

generated.

The inequality (57) then implies

S(Aµ ) ≥ 8π 2 |n|/g 2 .

The equality, which corresponds to a local minimum of the action, is obtained

for ¬elds satisfying the self-duality equations

Fµν = ±Fµν .

˜

These equations, unlike the general classical ¬eld equations (56), are ¬rst or-

der partial di¬erential equations and, thus, easier to solve. The one-instanton

solution, which depends on an arbitrary scale parameter », is

2 2xi

Ai = ’

Ai = (x4 δim + imk xk ) , m = 1, 2, 3 , . (63)

m 4

r2 + »2 r2 + »2

210 J. Zinn-Justin

The Semi-classical Vacuum. We now proceed in analogy with the analysis

of the CP (N ’ 1) model. In the temporal gauge A4 = 0, the classical minima

of the potential correspond to gauge ¬eld components Ai , i = 1, 2, 3, which are

pure gauge functions of the three space variables xi :

Am = ’ 1 iAm · σ = g(xi )‚m g’1 (xi ) .

2

The structure of the classical minima is related to the homotopy classes of map-

pings of the group elements g into compacti¬ed R3 (because g(x) goes to a

constant for |x| ’ ∞), that is again of S3 into S3 and thus the semi-classical

vacuum, as in the CP (N ’ 1) model, has a periodic structure. One veri¬es that

the instanton solution (63), transported into the temporal gauge by a gauge

transformation, connects minima with di¬erent winding numbers. Therefore, as

in the case of the CP (N ’ 1) model, to project onto a θ-vacuum, one adds a

term to the classical action of gauge theories:

iθ

Sθ (Aµ ) = S(Aµ ) + d4 x Fµν · Fµν ,

˜

2

32π

and then integrates over all ¬elds Aµ without restriction. At least in the semi-

classical approximation, the gauge theory thus depends on one additional pa-

rameter, the angle θ. For non-vanishing values of θ, the additional term violates

CP conservation and is at the origin of the strong CP violation problem: Except

if θ vanishes for some as yet unknown reason then, according to experimental

data, it can only be unnaturally small.

5.4 Fermions in an Instanton Background

We now apply this analysis to QCD, the theory of strong interactions, where NF

¯

Dirac fermions Q, Q, the quark ¬elds, are coupled to non-abelian gauge ¬elds

Aµ corresponding to the SU (3) colour group. We return here to standard SU (3)

notation with generators of the Lie Algebra and gauge ¬elds being represented

by anti-hermitian matrices. The action can then be written as

®

Nf

1

S(Aµ , Q, Q) = ’ d4 x ° 2 tr F2 + Qf (D + mf ) Qf » .

¯ ¯

µν

4g

f =1

The existence of abelian anomalies and instantons has several physical conse-

quences. We mention here two of them.

The Strong CP Problem. According to the analysis of Sect. 4.5, only con-

¬gurations with a non-vanishing index of the Dirac operator contribute to the

θ-term. Then, the Dirac operator has at least one vanishing eigenvalue. If one

fermion ¬eld is massless, the determinant resulting from the fermion integration

thus vanishes, the instantons do not contribute to the functional integral and

the strong CP violation problem is solved. However, such an hypothesis seems

to be inconsistent with experimental data on quark masses. Another scheme is

based on a scalar ¬eld, the axion, which unfortunately has remained, up to now,

experimentally invisible.

Chiral Anomalies and Topology 211

The Solution of the U (1) Problem. Experimentally it is observed that the

masses of a number of pseudo-scalar mesons are smaller or even much smaller

(in the case of pions) than the masses of the corresponding scalar mesons. This

strongly suggests that pseudo-scalar mesons are almost Goldstone bosons asso-

ciated with an approximate chiral symmetry realized in a phase of spontaneous

symmetry breaking. (When a continuous (non gauge) symmetry is spontaneously

broken, the spectrum of the theory exhibits massless scalar particles called Gold-

stone bosons.) This picture is con¬rmed by its many other phenomenological

consequences.

In the Standard Model, this approximate symmetry is viewed as the conse-

quence of the very small masses of the u and d quarks and the moderate value

of the strange s quark mass.

Indeed, in a theory in which the quarks are massless, the action has a chiral

U (NF ) — U (NF ) symmetry, in which NF is the number of ¬‚avours. The spon-

taneous breaking of chiral symmetry to its diagonal subgroup U (NF ) leads to

2

expect NF Goldstone bosons associated with all axial currents (corresponding

to the generators of U (N ) — U (N ) that do not belong to the remaining U (N )

symmetry group). In the physically relevant theory, the masses of quarks are

non-vanishing but small, and one expects this picture to survive approximately

with, instead of Goldstone bosons, light pseudo-scalar mesons.

However, the experimental mass pattern is consistent only with a slightly

broken SU (2) — SU (2) and more badly violated SU (3) — SU (3) symmetries.

From the preceding analysis, we know that the axial current corresponding

to the U (1) abelian subgroup has an anomaly. The WT identities, which imply

the existence of Goldstone bosons, correspond to constant group transformations

and, thus, involve only the space integral of the divergence of the current. Since

the anomaly is a total derivative, one might have expected the integral to vanish.

However, non-abelian gauge theories have con¬gurations that give non-vanishing

values of the form (40) to the space integral of the anomaly (36). For small

couplings, these con¬gurations are in the neighbourhood of instanton solutions

(as discussed in Sect. 5.3). This indicates (though no satisfactory calculation

of the instanton contribution has been performed yet) that for small, but non-

vanishing, quark masses the U (1) axial current is far from being conserved and,

therefore, no corresponding light almost Goldstone boson is generated.

Instanton contributions to the anomaly thus resolve a long standing experi-

mental puzzle.

Note that the usual derivation of WT identities involves only global chiral

transformations and, therefore, there is no need to introduce axial currents. In

the case of massive quarks, chiral symmetry is explicitly broken by soft mass

terms and WT identities involve insertions of the operators

Mf = mf ¯

d4 x Qf (x)γ5 Qf (x),

which are the variations of the mass terms in an in¬nitesimal chiral transforma-

tion. If the contributions of Mf vanish when mf ’ 0, as one would normally

expect, then a situation of approximate chiral symmetry is realized (in a sym-

212 J. Zinn-Justin

metric or spontaneously broken phase). However, if one integrates over fermions

¬rst, at ¬xed gauge ¬elds, one ¬nds (disconnected) contributions proportional

to

’1

Mf = mf tr γ5 (D + mf ) .

We have shown in Sect. 4.5) that, for topologically non-trivial gauge ¬eld con¬gu-

rations, D has zero eigenmodes, which for mf ’ 0 give the leading contributions

1

d4 x •— (x)γ5 •n (x)

Mf = mf + O(mf )

n

mf

n

= (n+ ’ n’ ) + O(mf ).

These contributions do not vanish for mf ’ 0 and are responsible, after inte-

gration over gauge ¬elds, of a violation of chiral symmetry.

6 Non-Abelian Anomaly

We ¬rst consider the problem of conservation of a general axial current in a

non-abelian vector gauge theory and, then, the issue of obstruction to gauge

invariance in chiral gauge theories.

6.1 General Axial Current

We now discuss the problem of the conservation of a general axial current in

the example of an action with N massless Dirac fermions in the background of

non-abelian vector gauge ¬elds. The corresponding action can be written as

¯ ¯

S(ψ, ψ; A) = ’ d4 x ψi (x)Dψi (x).

¯

In the absence of gauge ¬elds, the action S(ψ, ψ; 0) has a U (N )—U (N ) symmetry

corresponding to the transformations

+ γ5 )U+ + 1 (1 ’ γ5 )U’ ψ ,

1

ψ= 2 (1 2

+ γ5 )U† + 1 (1 ’ γ5 )U† ,

¯ ¯ 1

ψ =ψ 2 (1 (64)

’ +

2

where U± are N — N unitary matrices. We denote by t± the anti-hermitian

generators of U (N ):

U = 1 + θ± t± + O(θ2 ).

Vector currents correspond to the diagonal U (N ) subgroup of U (N ) — U (N ),

that is to transformations such that U+ = U’ as one veri¬es from (64). We

couple gauge ¬elds A± to all vector currents and de¬ne

µ

A µ = t ± A± .

µ

Chiral Anomalies and Topology 213

We de¬ne axial currents in terms of the in¬nitesimal space-dependent chiral

transformations

¯ ¯

U± = 1 ± θ± (x)t± + O(θ2 ) ’ δψ = θ± (x)γ5 t± ψ, δ ψ = θ± (x)ψγ5 t± .

The variation of the action then reads

¯

d4 x Jµ (x)‚µ θ± (x) + θ± (x)ψ(x)γ5 γµ [Aµ , t± ]ψ(x) ,

5±

δS =

5±

where Jµ (x) is the axial current:

¯

Jµ (x) = ψγ5 γµ t± ψ .

5±

Since the gauge group has a non-trivial intersection with the chiral group, the

commutator [Aµ , t± ] no longer vanishes. Instead,

[Aµ , t± ] = Aβ fβ±γ tγ ,

µ

where the fβ±γ are the totally antisymmetric structure constants of the Lie

algebra of U (N ). Thus,

d4 x θ± (x) ’‚µ Jµ (x) + fβ±γ Aβ (x)Jµ (x) .

5± 5γ

δS = µ

The classical current conservation equation is replaced by the gauge covariant

conservation equation

5±

Dµ Jµ = 0 ,

where we have de¬ned the covariant divergence of the current by

≡ ‚µ Jµ + f±βγ Aβ Jµ .

5 5± 5γ

Dµ Jµ µ

±

In the contribution to the anomaly, the terms quadratic in the gauge ¬elds are

modi¬ed, compared to the expression (36), only by the appearance of a new

geometric factor. Then the complete form of the anomaly is dictated by gauge

covariance. One ¬nds

1

D» J» (x) = ’

5±

tr t± Fµν Fρσ .

µνρσ

16π 2

This is the result for the most general chiral and gauge transformations. If we

restrict both groups in such a way that the gauge group has an empty intersection

with the chiral group, the anomaly becomes proportional to tr t± , where t± are

the generators of the chiral group G — G and is, therefore, di¬erent from zero

only for the abelian factors of G.

214 J. Zinn-Justin

6.2 Obstruction to Gauge Invariance