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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

+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

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 .
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

∆’1 (p) (1 + ip/Mr )(1 ’ ip/Mr ).
= (m + ip)

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).

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


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
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-

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

Z0 (A) = N det (m + D) 1 ’ D2 /Mr

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),


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).
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

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


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).

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
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
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 .

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

where the operator H is analogous to a non-relativistic hamiltonian in a magnetic
H = ’Dµ Dµ + m2 , H0 = ’∇2 + m2 .
UV divergences then arise from the small t integration. The integral over time


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