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 .

µ

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

µ

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

2

x,µ

The modi¬ed Dirac operator for the Fourier components of the ¬eld reads

i

(1 ’ cos apµ ) +

DW (p) = m + M γµ sin apµ . (17)

a

µ µ

The fermion propagator becomes

’1

† †

∆(p) = DW (p) DW (p)DW (p)

with

2

1

†

(1 ’ cos apµ ) sin2 apµ .

DW (p)DW (p) = m + M +

a2

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

Susskind.

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

1

ψ(x) = e’iΛ(x) ψ (x),

¯ ¯ Aµ (x) = ’ ‚ν Λ(x) + Aµ (x),

ψ(x) = eiΛ(x) ψ (x),

e

(19)

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),

5

5

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

¯

5

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) .

5

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

5

conclusion we now check by an explicit calculation of the expectation value of

5

‚µ Jµ (x) in the case of the action 18.

Remarks.

(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† ),

2

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

1234

integration by parts and anti-symmetrization shows that

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

µνρσ

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

5

‚µ Jµ (x), the divergence of the axial current. It is determined up to a multiplica-

tive constant:

‚» J» (x) ∝ e2 ∝ e2

5

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

5

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)

(2π)4

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-