PHYSICAL REViEW D VOLUME 47, NUMBER 8 15 APRIL 1993

Higher-derivative Schwinger model

R. L. P. G. Amaral, L. V. Belvedere, and N. A. Lemos

C. P. Natividade
Departamento de Matematica, Uni Uersidade Estadual Paulista, Campus de Guaratingueta, 12500 Sao Paulo, Sao Paulo, Brazil

(Received 23 October 1992)

Using the operator formalism, we obtain the bosonic representation for the free fermion field satisfy-
ing an equation of motion with higher-order derivatives. Then, we consider the operator solution of a
generalized Schwinger model with higher-derivative coupling. Since the increasing of the derivative or-
der implies the introduction of an equivalent number of extra fermionic degrees of freedom, the mass ac-
quired by the gauge field is bigger than the one for the standard two-dimensional QED. An analysis of
the problem from the functional integration point of view corroborates the findings of canonical quanti-
zation, and corrects certain results previously announced in the literature on the basis of Fujikawa s

technique.

PACS number(s): 11.10.Ef, 11.15.Tk

I. INTRODUCTION

Variational problems involving functionals that depend
on derivatives of order higher than the first appear to
have been first discussed by Ostrogradskii [1], who also
established the basis for the Hamiltonian treatment of
such problems. Although most physical systems are
characterized by Lagrangians that contain, at most, first
derivatives of the dynamical variables, there is a continu-
ing interest in the study of model field theories defined by
higher-derivative Lagrangians.

Early attempts to investigate higher-derivative theories
aimed at generalizing or amending certain physical
theories in order to get rid of some of their undesirable
properties. Along these lines Weyl and Eddington [2]
were, to the best of our knowledge, the first to add
curvature-squared terms to the Einstein-Hilbert Lagrang-
ian so as to extend the general theory of relativity.
Modifications to Maxwell's electromagnetic theory were
proposed by Bopp [3] and Podolsky [4] with the goal of
avoiding divergences such as the self-energy of a point
charge. Next Pais and Uhlenbeck [5] investigated wheth-
er the use of higher-order (including infinite-order) equa-
tions of motion might lead to the elimination of the diver-
gent quantities that plague quantum field theory. They
concluded that, in general, it is not possible to reconcile
finiteness, positivity of free field energy, and causality. In
other words, ghost states with negative norm and possi-
bly unitarity violation are unavoidable in higher-order
theories, and these facts became strong arguments against
such theories.

However, in spite of these shortcomings, higher-order
theories have never been entirely abandoned because they
also possess some good properties, justifying a sort of re-
vival of this subject in recent years. It has been shown [6]
that a quantum theory of gravitation constructed by add-
ing terms quadratic in the curvature to the Einstein-
Hilbert Lagrangian is asymptotically free and the prob-

lem of its renormalizability is attenuated. It must be em-
phasized that such curvature-squared terms show up nat-
urally as small corrections in the effective action of super-
string theories in the limit of zero slope [7]. Higher-
derivative terms appear naturally in the superfield formu-
lation of supersymmetric theories [8] and also occur in
the action proposed by Polyakov [9] in string theory,
which involves the extrinstic curvature of the world
sheet. It is further to be remarked that higher-order
corrections are very useful as a mechanism for regulariz-
ing ultraviolet divergences [10], especially of gauge-
invariant supersymmetric theories, since this is the only
available regularization method that preserves both
gauge invariance and supersymmetry [11].

Originally with functional methods, quantum and elec-
trodynamics in two spacetime dimensions with massless
fermions (Schwinger model) was exactly solved by
Schwinger [12] as an example of a theory in which gauge
invariance does not necessarily require a gauge field with
null physical mass. The physical content of the theory as
well as the correct interpretation of Schwinger's solution
became clearer with the appearance of the operator for-
mulation by Lowenstein and Swieca [13], in which the
fermion field is parametrized in terms of boson fields
("bosonization"), a method that had been previously em-

ployed by Klaiber [14] to study the Thirring model. The
boson representation of fermion fields turned out to be of
great utility for establishing several equivalences between
two-dimensional quantum field theories [15]. The
Schwinger model is an exactly soluble theory which ex-
hibits charge screening, fermion confinement, asymptotic
freedom, and a rich vacuum structure. This is why it
came to be regarded as a prototype model for
confinement of quarks. '

Recently, a modified version of the Schwinger model in
which fermion and gauge fields are coupled through
third-order derivatives was proposed [17] and studied by
functional methods. The axial anomaly and the dynami-

47 3443 1993 The American Physical Society



3AAA AMARAL, BELVEDERE, LEMOS, AND NATIVIDADE 47

cal mass generated for the photon field were calculated by
means of Fujikawa's path-integral technique [18],and the
results obtained [17] were the same as those for the stan-
dard Schwinger model.

The present work is devoted to the study of a general-
ized Schwinger model with higher derivatives, which in-
cludes as particular cases the original model and the one
recently proposed by Barcelos-Neto and Natividade [17].
First the free fermion theory is canonically quantized and
the exact expression for the n-point Wightman function is
derived. Then it is shown that the free theory is amen-
able to bosonization, and the representation of the fer-
mion field in terms of scalar and pseudoscalar fields is ex-
plicitly constructed. Next the generalized Schwinger
model is defined by requiring local gauge invariance of
the free fermion theory. The resulting Lagrangian densi-
ty exhibits an additional chiral gauge invariance at the
classical level. The quantum model, however, does not
display the same symmetry, and the anomalous diver-
gence of the axial-vector current is obtained. It is estab-
lished that the gauge field acquires a physical mass that
becomes larger as the order of the derivative of the fer-
mion field in the Lagrangian increases.

In the limit in which our theory reduces to the
Schwinger model the well-known usual results are
recovered. However, when it coincides with the theory
proposed by Barcelos-Neto and Natividade, our results
for the axial anomaly and dynamical generation of mass
for the gauge field differ from those found by the later au-
thors. This state of affairs prompted us to undertake a
reexamination of the problem by functional methods. An
analysis in terms of Fujikawa s path-integral technique
was carried out which corroborated our previous findings
in the framework of canonical quantization. Finally, the
reason for the discrepancy was identified and an error is
pointed out in the work of Barcelos-Neto and Nativi-
dade.

This paper is organized as follows. In Sec. II the gen-
eralized free fermion theory is introduced, and its canoni-
cal quantization and bosonization are performed. In Sec.
III the generalized Schwinger model is defined by
demanding local gauge invariance of the free theory. The
operator solution is found, the axial anomaly and the
physical mass acquired by the gauge field are derived, and
the vacuum structure is discussed. In Sec. IV the model
is investigated by Fujikawa s path-integral formalism and
the predictions of canonical quantization are confirmed.
Section V is dedicated to general comments and con-
clusions.

Our notation and conventions are as follows:

i)„=diag(1, —1), e '=1, [y",y I =2ii"

r"=r'r"'r' r"r =n" 1+~" r5

0 1

1 0
0 1

—1 0

—1 0
0 1

II. FREE THEORY: CANONICAL QUANTIZATION
AND BOSONIZATION

~ =g+ g&&+ ig +g* g2)v+ ig (2.2)

For the sake of simplicity and in order to show details
of the quantization procedure as well as some features of
the theory, we initially consider the case with third-order
(X = 1) derivatives.

Since in the case of free theory the two spinor com-
ponents decouple, we must treat them independently.
The upper spinor component will be considered first.

As the first step to quantize the theory we must obtain
the basic Poisson brackets. In the usual procedure for
higher-derivative theories [19], one would take g and its
two first time derivatives as independent variables
comprising the configuration space. In the present case,
previous experience in dealing with higher-derivative sca-
lar theories [20] suggests that we make a point transfor-
mation and take gii), ~3 gii), and 8 gii) as our basic vari-
ables. As will be seen, this choice of variables brings
much simpler expressions for the momenta. The varia-
tion of the action around a solution of the equation of
motion results in the following expressions for the respec-
tive conjugate momenta to the above variables:

II, = i 3 pi*, ),
II2= iB g(*))

II3=ig(*)) .

(2.3a)

(2.3b)

(2.3c)

From these canonical variables we obtain the following
nonvanishing equal-time anticommutation relations:

g,*„(~),g„,(y)I = —Ia g,*„(x),a g„,(y)I

Since we need the solution of the free theory in order
to obtain a full operator solution of the Schwinger model
with higher-order derivative couplings, in this section we
will consider in detail the free ease.

A generalization of the free theory to derivative of or-
der (2K+ 1) is given by the Lagrangian density

X,=g[W']~@, X=O, 1,2, 3, . . . , (2.1)

which exhibits global gauge and chiral symmetries. In-
troducing the two-dimensional spinor field g=(g~, ), g~z)),
the Lagrangian density (2.1) can be written as a sum of
two decoupled pieces:

rs=r'r'
x —=x'+x', O~=a, +0, .

= [P(*i)(~) ~ —0(i)(3') I

=6(x, —y) ) . (2.4)

The following explicit representation for the r ma-
trices is adopted:

It is worth remarking that, as is usually done, we could
have considered the g~ )

and g~* )
fields as independent

variables. In so doing, although the theory presents con-



HIGHER-DERIVATIVE SCHWINGER MODEL 3445

straints the resulting Dirac brackets would lead to the
same anticommutators as above.

In order to obtain the quantum solutions, let us intro-
duce the Fourier decomposition

k g(k, k+ )=0,
whose general solution is

g(k+, k ) =g, (k+ )5(k )+g~(k+ ) 6(k )
dk

(2.6)

g(, )(x)= Jdk+dk e + g(k+, k ) . (2.5)

Thus, the equation of motion in momentum space is
given by

+$3(k+ ) 6(k ) .
d2

dk
In this way, the upper spinor component (2.5) can be
written as

] +~
g(()(x) = J dk+ e

27T

x+
k)(k+ )+ 0z(k+ )+

2

$3(k+ ) e (2.8)

where m is a finite arbitrary mass scale introduced for
later convenience, and the g; fields have been redefined.
A convergence factor for large momenta has also been in-
troduced. The anticommutation relations now read

[g,(k ), g*(k' )}=—[g (k ), g*(k' )}
= g3(k+ ), g)*(k'+ )}=5(k+ —k+ ) .

l ~ (1)(k+ ) 0 (1)(k+ ) }
= [i)l (1)(k+ ) itl (1)(k+ ) }

[1 (1)(k+) 0(1)(k+ )}

=5(k+ —k'+ ) . (2.13)

Taking into account the Fourier representation for the
usual two-dimensional free Dirac field

(2.9)

) ~0) =g,*(—k ) ~0) =o, (2.10)

To construct the Fock space of the theory the vacuum
has to be defined. The annihilation operators will be
chosen through

1 + oo 1k+ x
P() )(x ) = dk+ e +

itt ~(()(k+ ),
277

the original g(x) field (2.8) may be written as
2

1 1+m x
v'2 m 2 2

(2.14)

for k+ )0 and i =1, 2, or 3. With such a choice the
one-particle states possess positive-definite energies.

The above-defined field (2.8) represents the operator
solution for the free theory described by the Lagrangian
(2.1), from which the Wightman functions of the theory
can be readily obtained. In particular, the two-point
function is then given by

(olg(()(x)g())(g)~0) =
+ +

l (p x )

16~ y+ —x++ie ' (2.11)

0('))(k+ ) = —I:P((k+ )+03(k+ ) }v'2 (2.12a)

0('))(k+)= ~- I:0((k+)—k(k+)}v'2

W())(k+)=k(k+) .

(2.12b)

(2.12c)

The commutation relations are now diagonalized and we
get

while higher-point functions are obtained by means of
Wick's theorem just as in the first-order case. Note that
the Wightman two-point function for the standard two-
dimensional free fermionic theory coincides with the
—(Ora ga $*~0) and (0~(a g)g" ~0) correlators.

In order to achieve a clear understanding of this model
and aiming at a bosonization scheme, we can further
redefine the fields in such a way as to obtain a diagonal
anticommutation structure. To this end, we define the
new left spinor component 1()( ) (a =Lorentz index)

r
1 1 m

v'2 m

+x P(i)(x )

2 2

(2.15)

J (x)=g(*()(x)a'g(„(x)—[a g(*))(x)}a g„)(x)

+ [a' g,*„(x)}g„,(x),
J (x)=g,*,(x)a g, ,(x)—[a g,*,(x)}a g, ,(x)

+ [ a + g(*p ) (x ) }g ( p ) ( x )

(2.16a)

(2.16b)

From the last expression, it is clear that the basic excita-
tions of the theory are three independent free fermionic
particles, one of them quantized with positive metric and
the other ones with negative metric.

It is to be stressed that, since the mapping (2.15) be-
tween the g and gl fields involves the time explicitly, the
total Hamiltonian, while expressed in terms of the diago-
nal 1' fields, does not have the usual form. Nevertheless,
the part of Hamiltonian that evolves the itl fields will be
the canonical one.

The expression for the lower spinor component can be
obtained from the upper one by simply making the re-
placement x +—~x +.

The conserved currents associated with the global
gauge and chiral symmetries are most easily obtained in
terms of the variables g, a g, and a g as independent
fields. In this way, we obtain the light-cone current com-
ponents



3446 AMARAL, BELVEDERE, LEMOS, AND NATIVIDADE

which satisfy independent conservation laws

(2.17)

In terms of the diagonal tti fermion field operators, the
current components (2.16) are given by

J (X ) = it)( ) )(X )g(1)(X) ill( 1)(X)g(1)(X )

1 (1)(x)1 ())(x) r

(x ) = it)(*2 )(x ) )tiI2 )( x ) iti(*2 ) (x ) iti( 2 )(x )

1)i(*2 )( X ) iti ( 2 ) (X )

(2.17a)

(2.17b)

In the expressions above, the minus signs together with
the fields quantized with negative metric ensure the
correct role for the generators of the global gauge and
chiral transformations. In terms of the diagonal fields,
the vector current can be written as

The higher-order derivative nature of the theory is re-
sponsible for the violation of clustering. For the upper
(lower) spinor component, the two-point function (2.20)
violates the cluster decomposition in the x (x+) light-
cone coordinate.

With the purpose of introducing the bosonization
scheme, let us consider the case N = 1 previously studied.
Since the P( )

fields are free and canonical, the bosoniza-
tion scheme to be employed is the standard one (see Hal-
pern, Ref. [15]). In order to ensure the correct anticom-
mutation relations (2.13), Klein factors must be intro-
duced. For the case X= 1, the bosonized expressions for
the free and canonical Dirac field operators P are given
by

1/2

%'~3:expIii/vr[y p, (x)+p)(x)]]:,

(2.21a)
3

(2.18)
1/2

APl Pl 3

2N+1
g() )(x ) = g f (x )1//() )(x .+ ) (2.19a)

with

f(x )= 1

2m

(mx /2) '
( —mx /2)+

(j—1)! (2N+1 —j)!
(2.19b)

and

1f2N+2 /(x-
2m

(mx /2)i
(j —1)!

/2)2N+ 1 —j
(2N+1 —j)!

where p indicates the positive or negative metric quanti-
zation prescription for the p fields (p 1

= —p 2
= —p 3

= 1).
The generalization to arbitrary (2N + 1) order is

straightforward. The upper component of the free fer-
mionic field can be written in terms of a set of (2N+ 1) di-
agonal fields as

X:exp ti i/~[y $2(x)+(52(x)]]:,
' 1/2

%'3:expIii/~[y' $3(x)+$3(x)]]:,

(2.21b)

In the expression (2.21) we have suppressed the Klein fac-
tors which ensure the anticommutation relations between
the two spinor components of P.

Making use of the bosonized expressions given by Eqs.
(2.21) and the standard point-splitting limit procedure
[21],we obtain, from (2.18),

1/2

g ()"P
1

(2.23)
7T j

Introducing the U(1) scalar potential @ via

(2.21c)

where (t . (P ) are free and massless scalar (pseudoscalar)
fields satisfying ()„i' =E„() i)(; p is an arbitrary finite
mass scale. The Klein factors%', are given by

%', =:exp i&~f B,it) (z)dz' (2.22)

for 1 ~j ~ N, whereas

(x /2)
+1

(2.19c)

(2.19d)

3

3 j —
1

we can write the vector current as

(2.24)

(2.25)

2
—(2N+1)

(
+ + )2N

(0~&*( )g (y)10&=
2N !vr x ——y

+—+i e
(2.20)

For j N the field is quantized with a positive metric, for
j (N +2) with a negative metric, and the metric sign of
the ((ii

+' fields is (
—1) .

For N odd (even) the diagonalization is performed in-
troducing N + 1 (N) fermionic fields quantized with nega-
tive (positive) metric and N (N + 1) fermionic fields quan-
tized with positive (negative) metric.

The generalized two-point function is given by

1
2

(I))
= —@+ g i(, . y

i =1
(2.26)

lDwhere the A, are two diagonal matrices of the SU(3). In
this way, the U(1) charge dependence can be factorized in
(2.21) and we get

We also introduce two independent canonical free scalar
lD

fields y such that the original fields can be written as
[22]



47 HIGHER-DERIVATIVE SCH%INGER MODEL 3447

[y 4(x)+N(x)) ".

3

X g f.(x )1~( )(x —
)

1/2
7r

P( )(x)=:exp.i

(2.27)

III. LOCAL GAUGE INVARIANCE

In this section we discuss the Schwinger model (SM)
with derivatives of order (2N + 1) defined by the Lagrang-
ian density

with
'(F——) +0'df, ' +"0'
4 pv (3.1)

1/2

X:exp i&vr+A, ,D(y5 y D+p'D)

where the covariant derivative of order (2N + 1) is
defined by

g(2N+ i) [ggyt)NI() (3.2)

the usual covariant derivative being given by

(2.28)
B=y"(8„ie—A„) . (3.3)

In the last expression we have suppressed the correspond-
ing Klein factors. Note that the I ~ operators are U(1)
neutral. Using the decomposition (2.26), the charges as-
sociated with the "component" fermions P can be writ-
ten as

)II( ) . (y5[p.x(x)+sg(x)].g( ) (3.4)

As in the usual (N =0) SM [13], the electromagnetic in-
teraction is introduced by performing a chiral operator
gauge transformation on the free fermionic field operator.
Thus, we write the operator solution as

O'= 0+e',

f '"a,e (z)dz',

where the U(1) charge reads
1/2

3

(2.29)

(2.30)

where g is the free fermionic field operator (2.15); A, and 6
are constants to be determined later on. The gauge field
is then identified as being given by

(3 5)

and the charges associated with the residual "infrafer-
mions" I J are

From Eq. (2.18), we see that the vector current can be
readily computed by the gauge-invariant point-splitting
limit prescription [13]

qJ=— (2.31) 2N+1
cf„=limZ '(e) g p [i)'jj(x+e)y„Q(x+e;x)P(x)

J =1
The generalization for arbitrary IV is straightforward, and
can be accomplished by introducing the 2N diagonal ma-
trices of the SU(2N + 1).

—VEV],

where VEV denotes the vacuum expectation value,

(3.6a)

Q(x+e;x)=exp i —y [AX(x+e)+5g(x+e)]+e f A„(z)dz"+y [AX(x)+6g(x)] (3.6b)

8"(x)= — XE" B,X(x)+L"(x),2' +1
7T

(3.7)

and Z is a finite renormalization constant. From (3.5)
and (2.21) we find

with X satisfying the equation of motion

(2N + 1)e ~( ) (3.10)

with
' 1/2

L„( )
(2N+1)5 ~„( )+ 2N+1 ~„@( )

(3.8)

Since the model possesses (2N+1) fermionic degrees of
freedom, the summation in Eq. (3.6) is responsible for the
appearance of the factor (2N + 1) in the expression (3.7)
for the current.

As in the usual case [13,21], due to the presence of the
longitudinal current L" the Maxwell equations are only
satisfied on the physical subspace defined by the subsidi-
ary condition

(3.9)

(2N+ 1)e
2vr

(3.11)

With the use of the decomposition (2.27), the fermion
field operator (3.4) can be written as

Condition (3.9) implies that L "(x), applied to the Fock
vacuum, generates states with zero norm. Hence, g must
be a canonical free field, quantized with indefinite metric.
This fixes the constant 6 to be t) =n/(2N+ 1). The con-
stant A, is chosen to be k=6 in order for 4 to approach
the free canonical fermion field at short distances. This
also ensures the canonical commutation relation for the
vector field 3".

The anomalous divergence of the axial-vector current
is then given by



3448 AMARAL, BELVEDERE, LEMOS, AND NATIVIDADE 47

4 (x)=:exp[i ny X(x)]::exp i

' 1/2 2N+1
[y5 [C(x)+g(x)]+@(x)]:g f (x )rj(x —

) . (3.12}

7T'P(x)~%'(x)=:exp i 2N+1 q(x) %(x),

The fermionic field operator which commutes with the
longitudinal current (3.8), thus belonging to the physical
subspace &~h„„ is obtained from the field (3.12) by per-
forming the following operator gauge transformation:

' 1/2

I

—(zN+1);(zN+1) &=~*,'~ '~++'lo& . (3.18)

The vacuum state above carries (2N+1) units of free
conserved charge and chirality. As in the standard SM,
the cluster decomposition is restored with respect to the
physical vacuum obtained in the usual way by consider-
ing the coherent superposition

A„(x)~A„(x)= A„(x)+-
e 2N+1

(3.13a)
1/2

8 q(x) .

(3.13b)

= 1 —in
1 9) —in202IO&&= g e ' ' ' 'In, ;n& .

n n
1 2

In this way we obtain

~.le„e, &
=e' .Ie, ; e, &,

(3.19)

(3.20)

4'(x)=:exp i 2N+1 y X(x)

2N+ 1

x g f (x+)rj(x+-)o. ,
j=1

(3.14}

where cr is an operator with a scale dimension of zero
given by

1/2

2N+1 [ y'[4(x)+ g(x) ]

+ [@(x)+g(x)]] (3.15)

The operator o. commutes with all the observables of the
theory. On the physical subspace defined by (3.9) it acts
as a constant operator which merely carries the bare
charge and chiral selection rules. As in the case of the
standard SM [13,21], an infinite set of vacuum states is
generated by repeated application of o on the Fock vacu-
urn. As is well known, this vacuum degeneracy implies a
violation of clustering.

This can be seen by considering the two-point func-
tions of operators carrying bare U(1) and chiral-U(1)
selection rules. Defining the operators 0 via

In this "physical" gauge, the fermionic field operator can
be factorized as

' 1/2

thus providing an irreducible representation for the ob-
servables.

IV. FUNCTIONAL INTEGRAL APPROACH

Our results for the anomalous divergence of the axial
current and the dynamical mass generated for the gauge
field, obtained in the framework of the operator solution,
appear to disagree with those derived previously [17]
through the use of Fujikawa's path-integral technique
[18]. Therefore, a reexamination of the problem in such
terms becomes a necessity.

The method introduced by Fujikawa to deal with
gauge theories with fermion fields rests on his observation
that although the classical Lagrangian is invariant under
a certain gauge transformation, the fermionic measure
(suitably defined) in the path integral may not be invari-
ant. If this is the case, the Jacobian of the transformation
induces additional terms in the Lagrangian which are re-
sponsible for anomalies and dynamical generation of
mass.

Let us consider the vacuum functional

Z=&J [dA„][dV][dq]e (4.1)

with JV a normalization factor and X given by Eq. (3.1)
with N= 1. The Lagrangian density X is invariant under
the infinitesimal transformations

2N+1
8 (x)= +:exp iy

j=1

1/2

r(x) :rj.(x)~.

(3.16)

%(x)= [1+icy e(x) ]'0'(x),

4(x)=%'(x)[1+icy e(x)] .

(4.2a)

(4.2b)

we obtain

lim &Ol@i (x)@2(x)8z(x+a )6,(x +a )IO&
~g~~~

=
I & oI @i(x)&(x) I

—(2N+1);(2N +1)& I'ao

(3.17)

In order to find out how the fermionic measure
changes, one first performs a Wick rotation to an Eu-
clidean spacetime by letting xo~ —ix4 and Ao~i A4.
Then

g=y (Bo—ieAO)+y'(8, —ieA, )

with ~y D4+y D1=&E ~

4 1 (4.3a)



47 HIGHER-DERIVATIVE SCHWINGER MODEL 3449

y—((3o i—eAo)+y'(8) —ieA, )

—y'D, +y'D, =(a'), , (4.3b)

where, formally,

I(x)=QQ„(x)y Q„(x) . (4.8)

)Ii(x) =ga„A„(x), (4.5a)

where y =I.y . Next one assumes that there exists an
orthonormal basis IQ„(x)J whose elements are eigen-
functions of the Hermitian operator II)E(II) )ERE, which
is the Euclidean version of the Dirac operator that occurs
in the fermionic part of the Lagrangian:

gE(g )E@EO„=A,„Q„. (4.4)

By expanding 4 and %' in terms of this basis, one has

This sum, however, is divergent and must be regularized.
Following Fujikawa's prescription, let us define I by

I= lim QQ„(x)y Q„(x)e
M~ oo

n
—[BE(,P )E8E ] f~= lim lim QQt(y)y'e Q„(x)

M~ oo y~x n

—[aE(a~)EaE)'IW' [2~lim lim Tr[y e e 5(2)(x —y)]M~ oo y~x

%(x)=QQ„(x)b„, (4.5b)
—Ikx»E[& ~)E&E] ~ ikx= lim Try e ' e e' ".

(2~)2

[d)Ii][d)Ii]=Qdb„da„.
n

(4.6)

According to standard computations [18], the change
undergone by the fermionic measure is given by

[d4][d4 ] = [d4 '] [d 4']exp 2ie f—d x e(x )I(x)

(4.7)

where a„b, are generators of an infinite-dimensional
Grassmann algebra. The expansions (4.5) possess the
property of diagonalizing the fermionic part of the ac-
tion, which justifies the following definition for the fer-
mionic measure in the path integral [18]:

(4.9)

0 p 0
D3 D3: () g (4 10)

+
[8E(8 )EBE]'=

0

where

D+ =ia4+D ) . (4.1 1)

Notice that the highest power of k occurring in the
operators P and Q applied to e'"" is —ks. Thus, having
taken the trace before integrating, it is convenient to
write Eq. (4.9) in the form

Making use of the explicit representation for the gamma
matrices one easily finds that

D+D

I= lim
M~ oo

=hm f
=hm f

—ikx —Q/M —P /M ikxd 2L-

2e e e
(2'�)

k kE/M;k —(g+ke )/M

(2m )

d'k kE™(k2)
(2vr)2

—(P+ ke )/M e'"

(4.12)

with

kE= —(k)+k4) . (4.13)

—idk&dk4 akEI= 1 4

(2') M

Expansion of the exponentials shows that g(kE) is of the
form

oo 27TK dK 4 ~6 I CX=—ia a e
(2~)2 1277

' (4.15)

k~ kE
g(kE) =a +P, +

M M' (4.14) —ikx(p g )
ikx —ikx(D3 D3 D3D3 )

ikx (4.16)

Now, a in Eq. (4.14) is given by the coefficient of kE in

where in each fraction with the denominator M, M',
and so on, only the term containing the highest power of
kE is displayed. Insertion of Eq. (4.14) into Eq. (4.12),
followed by the change of integration variable k ~Mk,
establishes that only the first term of the series (4.14)
gives a nonvanishing contribution to I in the limit
M~ ~. Thus

Considering only terms proportional to kE, by just count-
ing one finds (M = —ie A )

e '""(a++a+)(a++W+)(a++W+)(a +A )

&((9 +A )(Q +A )e'""=—9iekEB+ A

(4.17)



3450 AMARAL, BELVEDERE, LEMOS, AND NATIVIDADE 47

where symmetric terms under the exchange +~—have
been omitted. Therefore

leading to

[de][de]=[d% '][dq']
a= —9ie(B+ A —B A+ )

whence

38
(Minkowski) .

4~ ~

(4.18)

(4.19)

Xexp d x e(x)E„g„(x)(2N + 1)ie
27r

(4.28)

This last result leads finally to

[dV][d 4]= [d4 '] [d4']

Xexp f d x e(x)E g" (x) . (4.20)
3l8
2~

The variation of the fermionic part of the action under
the transformation (4.2) is

S~S+ief d x e(x)B„9~5(x),

where

(4.21)

8~ =3+y~y"8 @4+38 (4'y5gy y"4)
—a.a (ey y"y'y'e) . (4.22)

The anomalous axial divergence, defined by means of

5 =0,5e(x),=,
yields

(4.23)

(4.24)

Following the same line of reasoning as before we find
that the only nonvanishing contribution to I~ is

~ 2&K gK 4~ ~4N+2
I~ = ~+N K

(2~)'
ia&

2~(4N +2)

where, by counting,

(2N+1) e

(4.26)

(4.27)

As to dynamical generation of mass for the gauge field,
it can be discussed by a standard procedure [23].
Through successive infinitesimal steps one performs a
finite chiral gauge transformation that decouples photon
and fermion fields. With the repeated use of Eq. (4.20), at
the end of the calculation one reaches an action function-
al for a massless fermion field and a massive gauge field
without interaction between them. The mass acquired by
the photon turns out to be m =3e /m.

The above results for N = 1 can be easily extended to
arbitrary N. The regularized sum is now

d2k
IN= lim Try' 2eM~ oo (2m. )

Xexp( —[[g~(g )~] g~] /M +
) .

The same arguments used previously to find the
mass acquired by the photon now conduce to
m =(2N+1)e /n.

Our results obtained within the framework of the
path-integral formalism. coincide with those originally
found by means of the canonical quantization procedure,
but disagree with the findings of Barcelos-Neto and Na-
tividade [17], who also made use of Fujikawa's method.
The reason for the discrepancy is as follows. The opera-
tor that was used for regularizing the sum I in Ref. [17]
was Sz, instead of [Bz(P )EBz] . However, these two
operators are difFerent inasmuch as (B )z&(BE), that is,
the operations of taking the Hermitian conjugate and ro-
tating to Euclidean spacetime do not commute. As a
consequence, expansion of %' and %' in a basis of eigen-
functions of gz does not diagonalize the fermionic part
of the Euclidean action. But such a diagonalization is a
fundamental ingredient to justify Fujikawa's definition
(4.6) for the fermionic measure. As shown by Fujikawa,
his regularization prescription yields the same result
whether one chooses, for example, exp( Bz/M ) —or
exp( —gE /M ) as regularizing functions. Therefore,
what the authors of Ref. [17] really did was the computa-
tion of the chiral anomaly and of the mass acquired by
the photon for the standard Schwinger model. Un-
surprisingly, the usual results were obtained.

V. CONCLUSION AND OUTLOOK

The main result presented here was the proof that
higher-derivative generalizations of fermion models in
two spacetime dimensions are susceptible of bosoniza-
tion. This was shown both for a free theory and for a
generalized Schwinger model by explicitly constructing
the relevant Mandelstan operators. In the gauged ver-
sion the mass acquired by the gauge field becomes bigger
as the order of the highest derivative occurring in the La-
grangian increases. This can be quite naturally under-
stood as due to the increasing of the number of fermionic
degrees of freedom.

Although our results for the generalized Schwinger
model were first obtained within the cadre of canonical
quantization, their disagreement with previous results re-
ported on the basis of functional integration techniques
prompted us to investigate the model in the latter frarne-
work as well. Our findings could be reconciled with those
stemming from the application of Fujikawa's formalism
by choosing as regularization operator of the fermionic
Jacobian the one that is present in the fermionic part of
the action, a requirement indeed noted before in the
literature [24]. It should be noticed that the choice of
regularization operator in the Fujikawa scheme we em-
ployed was such as to preserve the classical symmetries of



47 HIGHER-DERIVATIVE SCHWINGER MODEL 3451

the action, but ultimately it was by comparison with the
reliable canonical approach that we fixed the scheme.

We also emphasize the amusing property that the bo-
sonization procedure has taken us from a higher-
derivative fermion theory to a bosonic multicomponent
model of first order. The point to be stressed is that
higher-derivative theories are plagued with undesired
properties such as nonunitarity and the existence of ghost
states. The absence of these difficulties in their bosonic
counterparts can be ascribed to the exponential operator
mapping from boson to fermion fields. It should be fur-
ther stressed the resemblance between the scheme
presented here and the Abelian bosonization of non-
Abelian fermionic models. In each of these models the
explicit symmetries that are present are not the same. In
the model treated here the boson counterpart exhibits an
O(2N+1) symmetry which is not explicit in the original

fermion model. Such a similarity leads us to wonder if in
our case there is an alternative scheme that preserves the
symmetries explicitly, like non-Abelian bosonization.

There are several interesting questions that can be
raised as continuation of the present work. For instance,
one may inquire about the possible bosonization of
higher-derivative generalizations of interacting fermion
theories such as the Thirring and Gross-Neveau massive
models. It also appears as promising the search for a not
at all unlikely discrete realization of the models investi-
gated here in condensed matter physics.

ACKNOWLEDGMENTS

The authors express their thanks to Conselho Nacional
de Desenvolvimento Cientifico e Tecnologico (CNPq),
Brazil, for partial finanical support.

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