Perturbative bosonization from two-point correlation functions

Perturbative bosonization from two-point correlation functions

D. Dalmazi,*A. de Souza Dutra,†and Marcelo Hott‡

UNESP, Campus de Guaratingueta´, DFQ, Av. Dr. Ariberto Pereira da Cunha, 333, CEP 12516-410, Guaratingueta´, SP, Brazil

共Received 10 January 2003; published 20 June 2003兲

Here we address the problem of bosonizing massive fermions without making expansions in the fermion masses in both massive QED2and QED3with N fermion flavors including also a Thirring coupling. We start

from two-point correlators involving the U(1) fermionic current and the gauge field. From the tensor structure of those correlators we prove that the U(1) current must be identically conserved共topological兲 in the corre-sponding bosonized theory in both D⫽2 and D⫽3 dimensions. We find an effective generating functional in terms of bosonic fields which reproduces these two-point correlators and from that we obtain a map of the Lagrangian density ␺¯r_{(i⳵”⫺m)␺}r _{into a bosonic one in both dimensions. This map is nonlocal but it is} independent of the electromagnetic and Thirring couplings, at least in the quadratic approximation for the fermionic determinant.

DOI: 10.1103/PhysRevD.67.125012 PACS number共s兲: 11.15.Bt

I. INTRODUCTION

One among the many dreams of theoretical physicists nowadays is the possibility of extending to higher dimen-sions (D⬎2) the bosonization of fermionic models. This can be justified by some good properties of one formulation as opposed to the other. For instance, strong coupling physics in one model corresponds to weak coupling in the other. A clas-sic example is the map between the massive Thirring and sine-Gordon models in D⫽2. Another interesting aspect of such a map is the fact that the usual electromagnetic charge in the fermionic model corresponds to the topological charge of the associated soliton field. Furthermore, we can have a map between a linear theory such as massive free fermions and a nonlinear one共sine-Gordon model at␤2_{⫽4␲) on the} other side.

In view of these and other interesting properties a lot of work has been devoted to the issue of bosonization 关1–3兴 共see also 关4兴兲. There have also been many attempts to gener-alize those ideas to higher dimensions 关5–22兴. For massive fermions in D⫽2 most of the methods are based on expan-sions around massless fields that are local conformal theo-ries. In D⫽3, although the case of massless free fermions can still be mapped into a bosonic theory共see 关8兴兲 this theory is nonlocal. In addition, the conformal group is finite in D ⫽3 and not so powerful as in D⫽2 which makes expansions around the massless case nontrivial. The other possibility is to employ functional methods. Once again the case of mass-less fermions is easier to deal with since the fermionic deter-minant can be exactly calculated for m⫽0. For massive fer-mions in D⫽2 a nontrivial Jacobian under chiral transformations plays a key role in deriving the sine-Gordon model 共see 关23–27兴兲. In D⫽3, chiral transformations play no role, and although some nonperturbative information is known关21兴 about the fermionic determinant we are basically

left with approximate methods like the one used in this work. On one hand, the method used here is inspired in the ap-proach carried out in关19兴, which is rather simple and based on two-point correlation functions. On the other hand, our bosonization rules depart from those in关19兴 in the sense that they are independent of the interactions. We extend the ap-proach of关19兴 by introducing an electromagnetic interaction and making use of the 1/N expansion, which allows us to go beyond the lowest order in the coupling constants. In the next section we introduce the model we are working with and obtain a general expression for the generating functional of the current and gauge field correlators. The expression is valid for arbitrary dimensions and depends on the vacuum polarization tensor. In Secs. III and IV we make the calcula-tions explicit in D⫽2 and D⫽3 dimensions, respectively. We first obtain in the fermionic theory the two-point current correlation functions also involving the gauge field, and then we write the current in terms of bosonic fields and derive the corresponding action for such fields that reproduces their correlators. In the final section we draw some conclusions and comment on similar approaches in the literature.

II. GENERATING FUNCTIONALS

We start by introducing the notation that will be used in both D⫽2 and D⫽3. The generating functional for a gen-eralized QED with Thirring self-interaction is given by

Z关J_␮r ,K_␮兴⫽

冕

DA_␮

兿

r⫽1 N D␺r D␺¯rexp

再

i

冕

d D_x

冋

_⫺1 4F␮␯ 2 ⫹␺¯_r

冉

_{i⳵”⫺m⫺} e

冑

NA”

冊

␺r⫺ g2 2N共␺¯r␥ ␮_␺ r兲2 ⫹␭ 2共⳵␮A ␮_兲2_⫹J ␮ r_共␺¯ r␥␮␺r兲⫹K␮A␮

册冎

, 共1兲

where N is the number of fermion flavors and summation over the repeated flavor index r (r⫽1,2, . . . ,N) is assumed. *Email address: dalmazi@feg.unesp.br

†_{Email address: dutra@feg.unesp.br} ‡_{Email address: hott@feg.unesp.br}

It is convenient to introduce an auxiliary vector field B_␮and work with the physically equivalent generating functional:

Z关J_␮r ,K_␮兴⫽

冕

DA_␮ DB_␮

兿

r⫽1 N D␺r D␺¯r ⫻exp

再

i

冕

dDx

冋

⫺1 4F␮␯ 2 _⫹1 2B␮B ␮ ⫹␭₂共⳵␮A␮兲2⫹␺¯r

冉

i⳵”⫺m⫺ e

冑

NA” ⫺ g

冑

NB” ⫹J” r

冊

_␺ r⫹K␮A␮

册冎

. 共2兲

Integrating over the fermionic fields we obtain

Z关J_␮r ,K_␮兴⫽

冕

DA_␮ DB_␮exp

再

i

冕

dnx

冋

⫺1 4F␮␯ 2 _⫹1 2B␮B ␮ ⫹2K␮A␮⫹ ␭ 2共⳵␮A ␮_兲2

册冎

⫻

兿

r⫽1 N det

冋

i⳵”⫺m⫺ 1

冑

N共eA” ⫹gB”兲⫹J” r

册

_{. 共3兲}

Now we compute the fermionic determinant, keeping terms of order (1/N)0 and (1/N)1/2. Higher order terms will be neglected. Furthermore, since we are interested only in two-point correlators, the terms higher than quadratic in the sources J_␮r will not be taken into account either. This amounts to the quadratic approximation for the fermionic determinant, which gives

Z关J_␮r ,K_␮兴⫽

冕

DA_␮DB_␮ exp ı 2

冕

dDk 共2␲兲D ⫻

再

⫺A˜␮关␪␮␯k2共1⫺␭兲⫹␭k2g␮␯兴A˜␯ ⫹B˜␮_B˜_␮⫹

兺

r_⫽1 N

冉

eA ˜ ␮

冑

N⫹g B ˜ ␮

冑

N⫺J˜␮ r

冊

⫻⌸␮␯_共k2_兲

冉

_eA˜␯

冑

N⫹g B ˜ ␯

冑

N⫺J˜␯ r

冊

_⫹2K˜ ␮A˜␮

冎

, 共4兲 where␪␮␯⫽g␮␯⫺k␮k␯/k2and the tildes over the fields rep-resent their Fourier transformations in momentum space. The quantity⌸␮␯ is the vacuum polarization tensor:

⌸␮␯_共k兲⫽i

冕

dDp 共2␲兲D tr

冋

1 p” ⫺m⫹i⑀␥ ␮ 1 共p” ⫹k”兲⫺m⫹i⑀␥␯

册

. 共5兲

In order to proceed further we have to calculate ⌸␮␯, which depends on the dimensionality of the space-time. III. BOSONIZATION FROM TWO-POINT CORRELATORS

IN DÄ2

In this section we restrict ourselves to the D⫽2 case. Using dimensional regularization we obtain, below the pair creation threshold (z⬅k2/4m2⬍1), ⌸␮␯⫽⌸˜ 共k2兲␪␮␯ 共6兲 with ⌸˜ 共k2_兲⫽1 ␲

冋

1⫺ 1 关z共1⫺z兲兴1/2arctan

冑

z 1⫺z

册

. 0⬍z⬍1, 共7兲 ⌸˜ 共k2_兲⫽1 ␲

冋

1⫺ 1 2 1

冑

z共z⫺1兲ln

冉

冑

共1⫺z兲⫹

冑

⫺z

冑

共1⫺z兲⫺

冑

⫺z

冊

册

, z⬍0. 共8兲 Once the tensor⌸␮␯is calculated one is left with a Gaussian integral over the vector fields A_␮ and B_␮ from which a gen-erating functional quadratic in the sources is derived. This generating functional furnishes the following two-point correlators:1

具

j_␮r共k兲j_␯s共⫺k兲

典

⫽⫺⌸ ˜2_共e2_⫺k2_g2_兲 ND ␪ ␮␯_⫹⌸˜_␪␮␯_␦rs_{, 共9兲}

具

j_␮r共k兲A_␯共⫺k兲

典

⫽ e⌸ ˜

冑

ND␪ ␮␯_{, 共10兲}

具

A␮共k兲A␯共⫺k兲

典

⫽

冉

⫺ 1 ␭k2⫹ 1⫹g2⌸˜ D

冊

␪ ␮␯_⫹g␮␯ ␭k2, 共11兲 where D⫽关⌸˜ e2⫺k2共1⫹g2⌸˜ 兲兴. 共12兲 The tensor structure of the above correlation functions is in full agreement with the corresponding Ward identities based on the U(1) symmetry, and it will play a key role in our bosonization procedure. At this point it is important to stress that our approach deviates from that of Ref. 关19兴. The 1/N expansion we have relied upon, which coincides with the quadratic approximation for the fermionic determinant, is not equivalent to the second order weak coupling expansion used in关19兴.

1_{More precisely, we should have written explicitly the two-point}

functions in the form具G(k)H( p)典⫽I(k)␦(2)_{(k⫹p) but in this}

ar-ticle we will not display the delta function as a matter of conve-nience. Notice also that when we write Ars⫽F⫹G␦rsit is assumed that F multiplies an N⫻N matrix where all entries are equal to 1.

Now, in order to derive a bosonized expression for the currents j_␮r⫽␺¯r␥␮␺r, we write down the most general

de-composition for a vector in the momentum space:

j_␤r共k兲⫽⑀_␤␦k␦␾r共k兲⫹k_␤␸r共k兲. 共13兲 Substituting it in Eq. 共9兲 and using the identity⑀_␤␦k␦⑀_␣␥k␥ ⫽k2_␪ ␣␤, we obtain

具

j_␣r共k兲j_␤s共⫺k兲

典

⫽⫺k2␪_␣␤

具

␾r共k兲␾s共⫺k兲

典

⫺k␣k␤

具

␸r共k兲␸s共⫺k兲

典

⫺⑀␣␦k␦k_␤

具

␾r共k兲␸s共⫺k兲

典

⫺⑀␤␥k␥k␣

具

␸r共k兲␾s共⫺k兲

典

⫽⫺⌸˜ 2_共e2_⫺k2_g2_兲 ND ␪␣␤⫹⌸ ˜_␪ ␣␤␦rs. 共14兲 From the above it is not difficult to derive

具

␸r_共k兲␸s_共⫺k兲

_典

_{⫽0, 共15兲}

具

␾r_共k兲␸s_共⫺k兲

_典

_{⫽0, 共16兲}

具

␾r_共k兲␾s_共⫺k兲

_典

_⫽⫺ 1 k2

冋

⫺ ⌸˜2_共e2_⫺k2_g2_兲 ND ⫹⌸ ˜_␦rs

册

_, 共17兲 from which one can safely set ␸⫽0. On the other hand, substituting the general decomposition

具

␾共k兲A␮共⫺k兲

典

⫽M苸␮␦k␦⫹Qk␮ 共18兲

in the mixed correlation functions

具

j_␮r(k)A_␯(⫺k)

典

given in Eq. 共10兲, we conclude that

具

␾r_共k兲A

␮共⫺k兲

典

⫽⫺

e⌸˜

冑

Nk2D⑀␮␦k

␦_{. 共19兲}

Now we are in a position to derive the bosonic Lagrang-ian densityL_B(A_␮,␾r_{) which is compatible with the}

corre-lation functions 共11兲, 共17兲, and 共19兲. For this purpose, we start from the following ansatz:

LB共A␮,␾r兲⫽␾rRrs␾s⫹2Sr␾r⑀␮␯k␯A␮

⫹A␮_A␯_共T

1␪␮␯⫹T2g␮␯兲, 共20兲 where R_rs, S_r, T₁, and T₂ are determined as follows. We introduce the external sources Xrand K_␮and define the gen-erating functional ZB关Xr,K␮兴⫽

冕

兿

r_⫽1 N D␾r _DA ␮exp

冉

i

冕

d2x关LB共A␮,␾r兲 ⫹Xr␾ r_⫹K ␮A␮兴

冊

. 共21兲

Assuming that Rrs is a symmetric nonsingular matrix, we

have performed the Gaussian integrals in Eq. 共21兲 and ob-tained an explicit formula for ZB关Xr,K␮兴 from which the

two-point correlators can be obtained. By matching these correlators with 共11兲, 共17兲, and 共19兲 we determine the bosonic Lagrangian uniquely:

LB共A␮,␾r兲⫽ 1 2

冋

r

兺

_⫽1 N ␾rk 2 ⌸˜ ␾ r_⫹g 2 Nk 2

冉

兺

r_⫽1 N ␾r

冊冉

兺

s_⫽1 N ␾s

冊

册

⫺A ␣_A␤ 2 关共1⫺␭兲k 2_␪ ␣␤⫹␭k2g␣␤兴 ⫹ e

冑

N⑀␮␯ A ␮_k␯

冉

兺

r⫽1 N ␾r

冊

_{. 共22兲}

All the steps that led us to Eq. 共22兲 are technically simple and not very elucidating. We should mention only that no expansions in either 1/N or any of the coupling constants have been made in those intermediate calculations. Further-more, notice that if we quantize LB(A␮,␾r) and integrate

over the scalar fields␾r in Eq.共22兲 this will lead to a non-local effective action for the photon, which was studied in 关28兴, where we concluded that, although nonlocal, the theory is free of tachyons. Next, by comparing with the Lagrangian density written in terms of fermionic fields in Eq. 共1兲 we have the bosonization formulas for each fermion flavor 共no sum over repeated roman indices below兲:

␺ ¯ r␥␮␺r共k兲⫽⑀␮␯k␯␾r共k兲, 共23兲 ⫺␺¯_{r共k兲共k”⫹m兲␺r共k兲⫽}1 2␾ rk 2 ⌸˜ ␾ r_{. 共24兲}

Now some comments are in order. First of all, if for some given flavor we do a U(1) transformation (␺r→ei␣␺r) in

the expectation value

具

j_␮r

典

and use any regularization scheme preserving the U(1) symmetry, it will be easy to derive the Ward identity

具

j_␮r⳵␯j_␯r

典

⫽0 which implies the tensor structure

具

j_␮r j_␯r

典

⬀␪_␮␯, and consequently we will have Eq. 共23兲. So the current is topological due to the U(1) global symmetry and that must hold nonperturbatively. On the other hand, the bosonization rule共24兲 is only approximate since the full ex-pression would require, in our approach, a complete knowl-edge of the fermionic determinant, which is possible only for m→0. In this case ⌸˜ →1/␲ and we end up with N massless scalar fields topologically coupled to the gauge field. Integra-tion over the gauge field leads to N⫺1 massless scalar modes and one mode with m2_⫽e2_/(N␲⫹g2_{), thus} repro-ducing the particular case of the so called Schwinger-Thirring model for N⫽1 关29兴, as well as the Schwinger model result m2⫽e2/␲ for g→0 and N⫽1. In the opposite limit of large mass (z→0) we have ␲˜→⫺2z/3 at leading order. Substituting back in Eq. 共22兲 we arrive at a divergent result at m→⬁ which is in agreement with the m→⬁ limit of the corresponding sine-Gordon model. See关30兴 for a

simi-lar comparison in the case of the massive Thirring model without electromagnetic coupling.

IV. BOSONIZATION FROM TWO-POINT CORRELATORS IN DÄ3

In D⫽3 dimensions the vacuum polarization tensor 共5兲 calculated by means of dimensional regularization is given by

⌸␮␯_{共k兲⫽i⌸}

1E␮␯⫹⌸2k2␪␮␯, 共25兲 with E␮␯⬅⑀␮␯␳k_␳ and, in the range 0⭐z⬍1,

⌸1⫽⫺ 1 8␲z1/2ln

冉

1⫹z1/2 1⫺z1/2

冊

, ⌸2⫽ 1 16␲mz

冋

1⫺

冉

1⫹z 2z1/2

冊

ln

冉

1⫹z1/2 1⫺z1/2

冊

册

, 共26兲 while for z⬍0 we have

⌸1⫽⫺ 1 4␲共⫺z兲1/2arctan

冑

⫺z, ⌸2⫽ 1 16␲mz

冋

1⫺

冉

1⫹z 2共⫺z兲1/2

冊

arctan

冑

⫺z

册

. 共27兲 In fact the ⌸₁ amounts to a regularization dependent finite term 关6,7兴, which was taken equal to zero due to the dimen-sional regularization used.

Substituting Eq.共25兲 in the general expression 共4兲 we can obtain the two-point functions

具

j_␮r共k兲j_␯s共⫺k兲

典

⫽⫺1 N

冋

k 2

冉

_⌸ 2⫹ P Q˜

冊

␪␮␯⫹i⌸1

冉

1⫺ k2 Q˜

冊

E␮␯

册

⫹共k2_⌸ 2␪␮␯⫹i⌸1E␮␯兲␦rs, 共28兲

具

j_␮r共k兲A_␯共⫺k兲

典

⫽ e Q˜

冑

N关P␪␮␯⫺i⌸1E␮␯兴, 共29兲

具

A_␮共k兲A_␯共⫺k兲

典

⫽g␮␯ ␭k2⫹

冋

e2P k2Q˜ ⫺ 共1⫹␭兲 ␭k2

册

␪␯␮⫺ ie2⌸1 k2Q˜ E␮␯, 共30兲 where we found it convenient to define

P⫽共e2⫺k2g2兲共k2⌸₂2⫺⌸₁2兲⫺k2⌸2, 共31兲 Q˜⫽k2_关共e2_⫺k2_g2_兲⌸

2⫺1兴2⫺共e2⫺k2g2兲2⌸1

2_{. 共32兲}

In analogy with the D⫽2 case, we use for the U(1) current a general decomposition in the momentum space

j_␣r共k兲⫽⑀_␣␤␥k␤Br␥共k兲⫹k␣␾r共k兲, 共33兲

from which we get

具

j_␣r共k兲j_␤s共⫺k兲

典

⫽⫺k_␣k_␤

具

␾r共k兲␾s共⫺k兲

典

⫹E␣␥E␦␤

具

Br␥共k兲Bs␦共⫺k兲

典

⫹k␣E␤␦

具

␾r共k兲Bs␦共⫺k兲

典

⫹k␤E␣␥

具

Br␥共k兲␾s共⫺k兲

典

. 共34兲

Multiplying the last expression by k␣k␤, we conclude that

具

␾r_(k)␾s_(⫺k)

_典

_{⫽0. Now, multiplying the resulting}

expres-sion by k␣, we have E_␣␤

具

␾r(k)B_s␤(⫺k)

典

⫽0. A similar ma-nipulation was used in the last section to derive Eqs. 共15兲, 共16兲, and 共17兲. Concluding, we can certainly neglect the sca-lar fields␾r⫽0 and minimally bosonize the U(1) current in D⫽3 in terms of a bosonic vector field. The bosonic version of the current is once again of topological nature and identi-cally conserved. As in the D⫽2 case, this happens because of the U(1) global symmetry of the fermionic Lagrangian. Taking ␾r⫽0 and substituting the decomposition 共33兲 in Eqs. 共28兲 and 共29兲, after some trivial manipulations we end up with

具

Br␥共k兲Bs␦共⫺k兲

典

⫽⫺Crsg␥␦⫹

冉

C⫺ H2 k2

冊

rs ␪␥␦_⫺共H1兲rs k2 E ␥␦_, 共35兲

具

A␯共k兲B_s␮共⫺k兲

典

⫽⫺

冉

ie⌸1

冑

NQ˜ ⫹D

冊

s g␯␮⫺

冉

e P

冑

NQ˜ k2

冊

s E␯␮ ⫹Ds␪␯␮, 共36兲

where Crs and Ds are arbitrary and

共H1兲rs⫽i⌸1␦rs⫹ i⌸1 N

冉

k2 Q˜ ⫺1

冊

, 共37兲 共H2兲rs⫽k2⌸2␦rs⫺ k2 N

冉

⌸2⫹ P Q˜

冊

. 共38兲 The arbitrariness of Crs and Ds shows that the bosonization

fields B_r␮ must be gauge fields, but the corresponding gauge symmetry is independent of the electromagnetic one. Analo-gous to the last section, we next derive a Lagrangian density compatible with the two-point correlators 共35兲, 共36兲, and 共30兲. Let us suppose we have a bosonic Lagrangian density

LB(A␮,Br␯) of the form LB共A␮,Br␯兲⫽Br␮O␮␯ rs_B s ␯_⫹us_B s ␮_E ␮␯A␯ ⫹A␮A␯共v1␪␮␯⫹v2g␮␯兲, 共39兲 whereO_␮␯rs has a general tensor structure in both momentum and flavor space,

O_␮␯rs_⫽ars_␪

The quantities ars,brs,drs,us,v1,v2 are determined by matching the two-point correlators derived from the generat-ing functional ZB关Y_␯ r ,K_␮兴⫽

冕

兿

r⫽1 N DBr␮ DA␮ exp

冉

i

冕

d 2_x关L B共A␮,Br␯兲 ⫹Y_␮r_B r ␮_⫹K ␮A␮兴

冊

, 共41兲

with the correlators 共35兲, 共36兲, and 共30兲. Assuming that the matrices ars,brs,drs are symmetric and nonsingular, our first step is to perform the Gaussian integrals over the N vector fields B_r␯, which furnishes

Z_B关Y_␯r,K_␮兴⫽

冕

DA_␮exp

再

i

冕

d2_x

冋

_⫺1 4共Yr ␮_⫹u rA␤E␤␮兲 ⫻共O⫺1_兲 ␮␯ rs_共Y s ␯_⫹u sE␯␤A␤兲 ⫹A␮_A␯_共v 1␪␮␯⫹v2g␮␯兲⫹K␮A␮

册冎

. 共42兲 The inverse operatorO⫺1has the same structure as Eq.共40兲: (O⫺1)_␮␯rs⫽a˜rs␪_␮␯⫹b˜rsE_␮␯⫹d˜rsg_␮␯, where the tilde vari-ables are functions of the nontilde ones. In particular, d˜rs

⫽(d⫺1₎

rs. Since we have the contractions E␮␯E␯␣⫽

⫺k2_␪

␣ ␮_{, E}␮␯_␪

␯␣⫽E␣␮, it becomes clear from Eq.共42兲, with

a little thought, that even after integrating over A_␮ the only term in log ZBquadratic in the sources Yr␮and contracted via

the metric tensor is ⫺1 4Yr␮Ys␯˜d

rs_g

␮␯. Therefore the

two-point functions 共35兲 require the identification Crs⫽(d˜/2)rs ⫽(d⫺1_/2)rs_.

Given that the matrix Crs is arbitrary, in order to simplify the calculations, we have chosen it proportional to the iden-tity, and consequently we reduce the undetermined param-eters in our original ansatz for the bosonic Lagrangian since drs⫽d␦rs. For analogous reasons, after integrating over A_␮, the crossed terms in the sources Y_r␮ and K␯ are contracted only by the tensors E_␮␯ and ␪_␮␯. Thus, the term propor-tional to g␯␮ on the right hand side of the correlation func-tions 共36兲 must vanish, which fixes D⫽⫺ie⌸1/

冑

NQ˜ . Fi-nally, comparing the other parts of the correlators共35兲, 共36兲, and共30兲 with the final expression for log ZB one can

deter-mine the quantities ars, brs, and usas functions of d, which remains unfixed. We spare the reader the lengthy details since they are totally technical and not very illuminating. We stress only that no further simplification hypothesis is used and no expansion is made in either 1/N or any of the cou-pling constants. In those intermediate steps all correlators are treated as if they were exact.

In order to make the final answer for the bosonic Lagrang-ian density more familiar, we have redefined the unfixed

pa-rameter d⬅⫺␭˜k2/2. In the momentum space we have

LB共A␮,Br␣兲⫽⫺ 1 2

冋

r

兺

_⫽1 N B_r␣共k2⌸2␪␣␤⫹i⌸1⑀␣␤␥k␥兲Br␤ k2⌸₂2⫺⌸₁2 ⫹g 2 N

冉

r

兺

⫽1 N B_r␣

冊

k2␪_␣␤

冉

兺

s⫽1 N B_s␤

冊

册

⫺A ␣_A␤ 2 关共1⫺␭兲k 2_␪ ␣␤⫹␭k2g␣␤兴 ⫹ e

冑

N⑀␮␯␥k ␥_A␮

兺

r⫽1 N B_r␯ ⫹

兺

r⫽1 N B_r␣B_r␤ 2 关␭˜k 2_␪ ␣␤⫺␭˜k2g␣␤兴. 共43兲

Notice that ␭ and ␭˜ are interpreted as independent gauge fixing parameters such that the generating functional 共41兲 reproduces the correlation functions共30兲, 共35兲, and 共36兲 with

Crs⫽⫺

1

␭˜k2␦rs, 共44兲 D⫽⫺ie⌸1

冑

NQ˜ . 共45兲

Integration over the vector fields B_r␣ will lead to an effective electromagnetic theory, which was also studied in 关28兴 共see also 关23兴兲, where we analyzed its pole structure and con-cluded that no tachyons appear, just as in its two-dimensional counterpart 共22兲.

Comparing Eq.共43兲 with the original Lagrangian 共1兲 writ-ten in terms of fermionic fields, we have the bosonization rules for each fermion flavor (r⫽1,2, . . . ,N):

␺ ¯_{r␥␮␺r共k兲⫽⑀␮␯␣k}␯_B r ␣_{共k兲, 共46兲} ⫺␺¯ r共k兲共k”⫹m兲␺r共k兲⫽⫺ 1 2Br ␣共k2⌸2␪␣␤⫹i⌸1E␣␤兲 k2⌸₂2⫺⌸₁2 Br ␤_. 共47兲 As in the D⫽2 case, we obtained Eq. 共46兲 from the fact that for each fermion flavor we have the tensor structure

具

j_␮ j_␯

典

⬀a␪_␮␯⫹bE_␮␯, which was derived by calculating the fermionic determinant up to quadratic order. However, this tensor structure must hold beyond that approximation 共non-perturbatively兲 as a consequence of the Ward identity

具

j_␮⳵␯j_␯

典

⫽0, which must be true also in D⫽3, by using a gauge invariant regularization. On the other hand, the bosonization rule 共47兲 holds as a consequence of the qua-dratic approximation for the fermionic determinant. Taking m→0 共or z→⫺⬁) in the expressions 共27兲 we have ⌸1→0 and ⌸2→1/(16

冑

⫺k2). Plugging these values back in Eqs. 共46兲 and 共47兲, we recover the result of 关8兴 with ␤⫽1/4 and ␪⫽0 in that reference, which deals with massless fermions.

Our rules共46兲 and 共47兲 also agree with the case of massive fermions treated in关15兴. We stress, however, that in both 关8兴 and 关15兴 only free fermions were considered, while our re-sults were derived for an interacting theory with Thirring and electromagnetic couplings. This indicates that our bosoniza-tion rules are rather universal at the quadratic level. In the opposite limit of heavy fermions (k2/4m2)⫽z→0 in Eq. 共26兲 we have, at leading order, ⌸2→0 and ⌸1→⫺1/(4␲) which leaves us 关see Eq. 共47兲兴, with a bosonic local and topological Chern-Simons Lagrangian for the field B_r␮, in agreement with the findings of 关10兴 共see also 关11兴 and 关12兴 for higher orders兲.

V. CONCLUSIONS

We derived bosonization maps for the U(1) currents and the fermion Lagrangian densities for both QED2 and QED3 with N fermion flavors and Thirring self-interaction. Both results hold for finite fermion masses and no derivative ex-pansion in k/m is made as in关31兴. By turning off the inter-actions we can reproduce the results of 关15兴 for a particular choice of the regularization parameter in that reference. Our calculations show that the U(1) currents, when written in terms of bosons, must be identically conserved共topological兲 as a direct consequence of the Ward identity

具

j_␮⳵␯j_␯

典

⫽0,

which is true nonperturbatively if there is no U(1) anomaly. In particular, there is no need to look at higher point func-tions or compute the fermionic determinant beyond the qua-dratic approximation to confirm that. On the other hand, our maps for the Lagrangian densities 共24兲 and 共47兲, which are also independent of the interactions, hold only perturbatively due to our quadratic approximation for the fermionic deter-minant, which amounts to neglecting terms of order higher than (1/N)1/2in the fermionic determinant and considering at most two-point current correlators to derive the bosonization rules.

Finally, we should mention that the results derived here could have been obtained in a technically more direct way along the lines of关20兴 共see also 关32兴兲 but the approach used here does not use any auxiliary field and clarifies the funda-mental role of the two-point correlators. It might also be useful 共work in progress兲 for deriving approximate bosonic maps for other fermion bilinears, like the mass term ␺¯␺ in D⫽3, which apparently could not be done using the ap-proach of 关20兴.

ACKNOWLEDGMENTS

This work was partially supported by CNPq and FAPESP, Brazilian research agencies.

关1兴 S. Coleman, R. Jackiw, and L. Susskind, Ann. Phys. 共N.Y.兲 93,

267共1975兲.

关2兴 S. Coleman, Phys. Rev. D 11, 2088 共1975兲; Ann. Phys. 共N.Y.兲 101, 239共1976兲.

关3兴 S. Mandelstam, Phys. Rev. D 11, 3026 共1975兲.

关4兴 E. Abdalla, M.C. Abdalla, and K.D. Rothe, Non-Perturbative

Methods in Two Dimensional Quantum Field Theory 共World Scientific, Singapore, 1991兲.

关5兴 A.M. Polyakov, in Field Theory and Critical Phenomena,

1988 Les Houches Lectures, edited by E. Bre´zin and J. Zinn-Justin共Elsevier, Amsterdam, 1989兲.

关6兴 A. Coste and M. Luscher, Nucl. Phys. B323, 631 共1989兲. 关7兴 M. Luscher, Nucl. Phys. B326, 557 共1989兲.

关8兴 E.C. Marino, Phys. Lett. B 263, 63 共1991兲.

关9兴 C.P. Burgess, C. Lutken, and F. Quevedo, Phys. Lett. B 336, 18 共1994兲.

关10兴 E. Fradkin and F.A. Schaposnik, Phys. Lett. B 338, 253 共1994兲; G. Rossini and F.A. Schaposnik, ibid. 338, 465 共1994兲. 关11兴 R. Banerjee, Phys. Lett. B 358, 297 共1995兲.

关12兴 R. Banerjee, Nucl. Phys. B465, 157 共1996兲. 关13兴 F.A. Schaposnik, Phys. Lett. B 356, 39 共1995兲.

关14兴 N. Bralic, E. Fradkin, V. Manias, and F.A. Schaposnik, Nucl.

Phys. B446, 144共1995兲.

关15兴 D.G. Barci, C.D. Fosco, and L.E. Oxman, Phys. Lett. B 375,

367共1996兲.

关16兴 N. Banerjee, R. Banerjee, and S. Ghosh, Nucl. Phys. B481,

421共1996兲.

关17兴 J.C. Le Guillou, C. Nu´n˜ez, and F.A. Schaposnik, Ann. Phys. 共N.Y.兲 251, 426 共1996兲.

关18兴 R. Banerjee and E.C. Marino, Phys. Rev. D 56, 3763 共1997兲. 关19兴 R. Banerjee and E.C. Marino, Nucl. Phys. B507, 501 共1997兲. 关20兴 D.G. Barci, L.E. Oxman, and S.P. Sorella, Phys. Rev. D 59,

105012共1999兲.

关21兴 D.G. Barci, V.E.R. Lemes, C. Linhares de Jesus, M.B.D. Silva

M. Porto, S.P. Sorella, and L.C.Q. Vilar, Nucl. Phys. B524, 765

共1998兲.

关22兴 A. de Souza Dutra and C.P. Natividade, Mod. Phys. Lett. A 14,

307共1999兲; Phys. Rev. D 61, 027701 共2000兲.

关23兴 D. Dalmazi, A. de Souza Dutra, and M. Hott, Phys. Rev. D 61,

125018共2000兲.

关24兴 K. Furuya, R.E. Gamboa-Saravi, and F.A. Schaposnik, Nucl.

Phys. B208, 159共1982兲.

关25兴 C.M. Naon, Phys. Rev. D 31, 2035 共1985兲. 关26兴 L.C.L. Botelho, Phys. Rev. D 33, 1195 共1986兲.

关27兴 P.H. Damgaard, H.B. Nielsen, and R. Sollacher, Nucl. Phys. B385, 227共1992兲; B296, 132 共1992兲.

关28兴 E.M.C. Abreu, D. Dalmazi, A. de Souza Dutra, and M. Hott,

Phys. Rev. D 65, 125030共2002兲.

关29兴 A. de Souza Dutra, C.P. Natividade, H. Boschi-Filho, R.L.P.G.

Amaral, and L.V. Belvedere, Phys. Rev. D 55, 4931共1997兲.

关30兴 K-I. Kondo, Prog. Theor. Phys. 94, 899 共1995兲.

关31兴 S. Ghosh, ‘‘Bosonization and Duality in 2⫹1 Dimensions:

Ap-plications in Gauged Massive Thirring Model,’’

hep-th/9901051.