VI (φ) ≡ dd x VI φ(x) .

None of the momentum cut-o¬ regularizations described so far can deal with

the determinant. As long as the determinant is a divergent constant that cancels

in normalized correlation functions, this is not a problem, but in the case of

a determinant in the background of an external ¬eld (which generates a set of

one-loop diagrams) this may become a serious issue.

(ii) This problem is related to another one: even in simple quantum mechan-

ics, some models have divergences or ambiguities due to problem of the order

between quantum operators in products of position and momentum variables. A

class of Feynman diagrams then cannot be regularized by this method. Quan-

tum ¬eld theories where this problem occurs include models with non-linearly

realized (like in the non-linear σ model) or gauge symmetries.

Chiral Anomalies and Topology 173

Global Linear Symmetries. To implement symmetries of the classical ac-

tion in the quantum theory, we need a regularization scheme that preserves the

symmetry. This requires some care but can always be achieved for linear global

symmetries, that is symmetries that correspond to transformations of the ¬elds

of the form

φR (x) = R φ(x) ,

where R is a constant matrix. The main reason is that in the quantum hamilto-

nian ¬eld operators and conjugate momenta are not mixed by the transformation

and, therefore, the order of operators is to some extent irrelevant. To take an

example directly relevant here, a theory with massless fermions may, in four

dimensions, have a chiral symmetry

¯ ¯

ψθ (x) = eiθγ5 ψ(x), ψθ (x) = ψ(x)eiθγ5 .

The substitution (5 )(for m = 0) preserves chiral symmetry. Note the importance

here of being able to work at ¬xed dimension four because chiral symmetry is

de¬ned only in even dimensions. In particular, the invariance of the integration

¯

measure [dψ(x)dψ(x)] relies on the property that tr γ5 = 0.

2.2 Regulator Fields

Regularization in the form (2) or (5) has another equivalent formulation based

on the introduction of regulator ¬elds. Note, again, that some of the regulator

¬elds have unphysical properties; for instance, they violate the spin“statistics

connection. The regularized quantum ¬eld theory is physically consistent only

for momenta much smaller than the masses of the regulator ¬elds.

Scalar Fields. In the case of scalar ¬elds, to regularize the action (1) for the

scalar ¬eld φ, one introduces additional dynamical ¬elds φr , r = 1, . . . , rmax , and

considers the modi¬ed action

1 1

Sreg. (φ, φr ) = φ ’∇2 + m2 φ + φr ’∇2 + Mr φr

dd x 2

2 2zr

r

+VI (φ + r φr ) . (7)

With the action 7 any internal φ propagator is replaced by the sum of the φ

propagator and all the φr propagators zr /(p2 + Mr ). For an appropriate choice

2

of the constants zr , after integration over the regulator ¬elds, the form (2) is

recovered. Note that the condition of cancellation of the 1/p2 contribution at

large momentum implies

1+ zr = 0 .

r

Therefore, not all zr can be positive and, thus, the ¬elds φr , corresponding to

the negative values, necessarily are unphysical. In particular, in the integral over

these ¬elds, one must integrate over imaginary values.

174 J. Zinn-Justin

Fermions. The fermion inverse propagator (5) can be written as

rmax

∆’1 (p) (1 + ip/Mr )(1 ’ ip/Mr ).

= (m + ip)

F

r=1

This indicates that, again, the same form can be obtained by a set of regulator

¯

¬elds {ψr± , ψr± }. One replaces the kinetic part of the action by

¯ ¯

dd x ψ(x)(‚ + m)ψ(x) ’ dd x ψ(x)(‚ + m)ψ(x)

1 ¯

dd x ψr (x)(‚ + Mr )ψr (x).

+

zr

=±,r

¯

Moreover, in the interaction term the ¬elds ψ and ψ are replaced by the sums

¯ ¯ ¯

ψ’ψ+ ψ’ψ+

ψr , ψr .

r, r,

For a proper choice of the constants zr , after integration over the regulator ¬elds,

the form (5) is recovered.

For m = 0, the propagator (5) is chiral invariant. Chiral transformations

change the sign of mass terms. Here, chiral symmetry can be maintained only if,

in addition to normal chiral transformations, ψr,+ and ψ’r are exchanged (which

implies zr+ = zr’ ). Thus, chiral symmetry is preserved by the regularization,

even though the regulators are massive, by fermion doubling. The fermions ψ+

¯ ¯¯

and ψ’ are chiral partners. For a pair ψ ≡ (ψ+ , ψ’ ), ψ ≡ (ψ+ , ψ’ ), the action

can be written as

¯

dd x ψ(x) (‚ — 1 + M 1 — σ3 ) ψ(x),

where the ¬rst matrix 1 and the Pauli matrix σ3 act in ± space. The spinors

then transform like

ψθ (x) = eiθγ5 —σ1 ψ(x), ψθ (x) = ψ(x)eiθγ5 —σ1 ,

¯ ¯

because σ1 anticommutes with σ3 .

2.3 Abelian Gauge Theory

The problem of matter in presence of a gauge ¬eld can be decomposed into two

steps, ¬rst matter in an external gauge ¬eld, and then the integration over the

gauge ¬eld. For gauge ¬elds, we choose a covariant gauge, in such a way that

power counting is the same as for scalar ¬elds.

Chiral Anomalies and Topology 175

Charged Fermions in a Gauge Background. The new problem that arises

in presence of a gauge ¬eld is that only covariant derivatives are allowed be-

cause gauge invariance is essential for the physical consistency of the theory.

The regularized action in a gauge background now reads

¯ ¯

S(ψ, ψ, A) = 1 ’ D2 /Mr ψ(x),

dd x ψ(x) (m + D) 2

r

where Dµ is the covariant derivative

Dµ = ‚µ + ieAµ .

Note that up to this point the regularization, unlike dimensional or lattice reg-

ularizations, preserves a possible chiral symmetry for m = 0.

The higher order covariant derivatives, however, generate new, more singular,

gauge interactions and it is no longer clear whether the theory can be rendered

¬nite.

Fermion correlation functions in the gauge background are generated by

¯

Z(¯, ·; A) =

· dψ(x)dψ(x)

¯ ¯

— exp ’S(ψ, ψ, A) + dd x · (x)ψ(x) + ψ(x)·(x)

¯ , (8)

where · , · are Grassmann sources. Integrating over fermions explicitly, one ob-

¯

tains

Z(¯, ·; A) = Z0 (A) exp ’ dd x dd y · (y)∆F (A; y, x)·(x) ,

· ¯

Z0 (A) = N det (m + D) 1 ’ D2 /Mr

2

,

r

where N is a gauge ¬eld-independent normalization ensuring Z0 (0) = 1 and

∆F (A; y, x) the fermion propagator in an external gauge ¬eld.

Diagrams constructed from ∆F (A; y, x) belong to loops with gauge ¬eld prop-

agators and, therefore, can be rendered ¬nite if the gauge ¬eld propagator can

be improved, a condition that we check below. The other problem involves the

determinant, which generates closed fermion loops in a gauge background. Using

ln det = tr ln, one ¬nds

ln Z0 (A) = tr ln (m + D) + tr ln 1 ’ D2 /Mr ’ (A = 0),

2

r

or, using the anticommutation of γ5 with D,

det(D + m) = det γ5 (D + m)γ5 = det(m ’ D),

ln Z0 (A) = tr ln m2 ’ D2 + tr ln 1 ’ D2 /Mr ’ (A = 0).

2

1

2

r

176 J. Zinn-Justin

One sees that the regularization has no e¬ect, from the point of view of power

counting, on the determinant because all contributions add. The determinant

generates one-loop diagrams of the form of closed fermion loops with external

gauge ¬elds, which therefore require an additional regularization.

As an illustration, Fig. 1 displays on the ¬rst line two Feynman diagrams

involving only ∆F (A; y, x), and on the second line two diagrams involving the

determinant.

The Fermion Determinant. Finally, the fermion determinant can be regular-

ized by adding to the action a boson regulator ¬eld with fermion spin (unphysical

since violating the spin“statisitics connection) and, therefore, a propagator sim-

ilar to ∆F but with di¬erent masses:

¯ ¯

SB (φ, φ; A) = 1 ’ D2 /(Mr )2 φ(x).

dd x φ(x) M0 + D

B B

r=1

¯

The integration over the boson ghost ¬elds φ, φ adds to ln Z0 the quantity

δ ln Z0 (A) = ’ 1 tr ln (M0 )2 ’ D2 ’ tr ln 1 ’ D2 /(Mr )2 ’ (A = 0).

B B

2

r=1

Expanding the sum ln Z0 + δ ln Z0 in inverse powers of D, one adjusts the masses

to cancel as many powers of D as possible. However, the unpaired initial fermion

mass m is the source of a problem. The corresponding determinant can only be

B

regularized with an unpaired boson M0 . In the chiral limit m = 0, two options

are available: either one gives a chiral charge to the boson ¬eld and the mass

B

M0 breaks chiral symmetry, or one leaves it invariant in a chiral transformation.

In the latter case one ¬nds the determinant of the transformed operator

eiθ(x)γ5Deiθ(x)γ5 (D + M0 )’1 .

B

Fig. 1. Gauge“fermion diagrams (the fermions and gauge ¬elds correspond to contin-

uous and dotted lines, respectively).

Chiral Anomalies and Topology 177

For θ(x) constant eiθγ5D = De’iθγ5 and the θ-dependence cancels. Otherwise a

non-trivial contribution remains. The method thus suggests possible di¬culties

with space-dependent chiral transformations.

Actually, since the problem reduces to the study of a determinant in an

external background, one can study it directly, as we will starting with in Sect. 4.

One examines whether it is possible to de¬ne some regularized form in a way

consistent with chiral symmetry. When this is possible, one then inserts the one-

loop renormalized diagrams in the general diagrams regularized by the preceding

cut-o¬ methods.

The Boson Determinant in a Gauge Background. The boson determinant

can be regularized by introducing a massive spinless charged fermion (again

unphysical since violating the spin“statisitics connection). Alternatively, it can

be expressed in terms of the statistical operator using Schwinger™s representation

(tr ln = ln det)

∞

dt ’tH0

’ e’tH ,

ln det H ’ ln det H0 = tr e

t

0

where the operator H is analogous to a non-relativistic hamiltonian in a magnetic

¬eld,

H = ’Dµ Dµ + m2 , H0 = ’∇2 + m2 .

UV divergences then arise from the small t integration. The integral over time