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%TCIDATA{Created=Thu Sep 19 17:39:41 2002}
%TCIDATA{LastRevised=Thu Dec 19 17:19:25 2002}
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\begin{document}

\title{\textbf{Perturbative bosonization from two-point correlation functions}}
\author{D. Dalmazi, A. de Souza Dutra and Marcelo Hott \\
%EndAName
\textit{{UNESP - Campus de Guaratinguet\'a - DFQ} }\\
\textit{{Av. Dr. Ariberto Pereira da Cunha, 333} }\\
\textit{{CEP 12516-410 - Guaratinguet\'a - SP - Brazil.} }\\
\textsf{E-mail: dalmazi@feg.unesp.br, dutra@feg.unesp.br }\\
\textsf{and hott@feg.unesp.br}}
\date{\today}
\maketitle

\begin{abstract}
Here we address the problem of  bosonizing massive fermions without making expansions
in the fermion masses in both massive $QED_2$ and $QED_3$ with $\, 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
(topological)  in the corresponding bosonized theory both in $D=2$ and $D=3$
dimensions. We find an effective bosonic action which reproduces those two point
correlators and from such action we obtain a map for the Lagrangian density
$\bar{\psi}^{r}\,(i\,\partial \!\!\!/\,-\,m\,){\psi}^{r}$  in terms of the
corresponding bosonic field in both dimensions. This map is nonlocal but it is
 independent of the eletromagnetic and
Thirring couplings, at least in the quadratic approximation for the fermionic
determinant.

\textit{{PACS-No.:} 11.15.Bt , 11.15.-q }
\end{abstract}

%\begin{Large}

%\baselineskip=20pt

\newpage

\section{Introduction}

%\begin{large}

One among many dreams of the theoretical physicists nowadays is the possibility of
extending to higher dimensions ($D>2$) the bosonization of fermionic models. This can
be justified by some good properties of one formulation  in contrast with the other
one. For instance, strong coupling physics in one model corresponds to weak coupling
in the other one. A classic example is the map between the massive Thirring and
Sine-Gordon models in $D=2$.  Another interesting aspect of such 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 like massive free fermions and a nonlinear one
 (Sine-Gordon at $\beta ^{2}=4\,\pi $) on the other side.


In view of these and other interesting properties a lot of work has been devoted to
the issue of bosonization \cite {Jackiw,Coleman,mandelstam} ( see also
\cite{livroElcio} ). There has also been many attempts to generalize those ideas to
higher dimensions [5-18]. For massive fermions in $\, D=2 \, $ most of the methods are
 based on expansions around massless fields which are
local conformal theories. In $D=3$, although the case of massless free fermions can
still be mapped into a bosonic theory ( see \cite{marino} ) such theory is nonlocal.
Besides, 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 massless fermions
is easier to deal with since the fermionic determinant can be exactly calculated for
$m=0$. For massive fermions in $D=2$ a nontrivial Jacobian under chiral
transformations plays a key role in deriving the Sine-Gordon model ( see [22-25]). In
$\, D=3\, $, chiral transformations play no role and although some nonperturbative
information is known \cite{linhares} about the fermionic determinant we are basically
left with approximate methods like the one used in this work. In the next section we
introduce the notation and obtain a general expression for the generating funtional of
the current and gauge field correlators. The expression is valid for arbitrary
dimensions and depends on the vacuum polarization tensor. In sections 3 and 4 we make
the calculations explicit in $D=2$ and $D=3$ dimensions respectively. We first obtain
in the fermionic theory the two point current correlation functions involving also the
gauge field and then we write the current in terms of bosonic fields and derive the
corresponding action for such fields which reproduces their correlators. In the final
section we draw some conclusions and comment on similar approaches in the literature.














\section{Generating Functionals }

We start by introducing the notation which will be used in both $D=2$ and $%
D=3$. The generating functional for a generalized QED
with Thirring self-interaction is given by
\begin{eqnarray}
Z\left[ J^r_{\mu },K_{\mu }\right] &=&\int \mathcal{D}A_{\mu }\,\prod_{r=1}^{N}\mathcal{D}%
\psi_r \,\mathcal{D}\bar{\psi}_r\,exp\left\{ i\,\int \,d^{D}x\,\,\left[ -\frac{1%
}{4}\,F_{\mu \nu }^{2}\,+\,\bar{\psi}_{r}\,(i\,\partial \!\!\!/\,-\,m\,-\,%
\frac{e}{\sqrt{N}}\,\pa)\psi_{r}\right. \right.  \nonumber \\
&&\left. \left. -\,\frac{g^{2}}{2N}\,(\,\overline{\psi }_{r}\gamma ^{\mu }\psi
_{r})^{2}\,\right] +\frac{\lambda }{2}\left( \partial _{\mu }A^{\mu }\right)
^{2}+\,J_{\mu }^{r}\left( \overline{\psi }_{r}\gamma ^{\mu }\psi_{r}\right) +\,K_{\mu
}\,A^{\mu }\right\} \ ,
\end{eqnarray}

\noindent where $N$ is the number of fermion flavors and summation over the
repeated flavor index $r$ ($r=1,2,...,N$) is assumed. It is convenient to
introduce an auxiliary vector field $B_{\mu }$ and work with the physically
equivalent generating functional:

\begin{eqnarray}
Z\left[ J^r_{\mu },K_{\mu }\right]  &=&\int \mathcal{D}A_{\mu }\,\mathcal{D}B_{\mu }
\prod_{r=1}^N\mathcal{D}\psi_r \,\mathcal{%
D}\bar{\psi}_r\,exp\left\{ \,i\,\int \,d^{D}x\,\,\left[ -\frac{1}{4}\,F_{\mu
\nu }^{2}\,+\,{\frac{1}{2}}B_{\mu }B^{\mu }+\right. \,\right.  \nonumber \\
&&\left. \left. +\bar{\psi}_r\,(i\,\partial \!\!\!/\,-\,m\,-\,\frac{e}{%
\sqrt{N}}\,\pa-\,\frac{g}{\sqrt{N}}\,\pb+J\!\!\!/^{r})\psi_r+\,K_{\mu }\,A^{\mu
}\right] \right\} \ ,
\end{eqnarray}

\no Integrating over the fermionic fields we obtain

\begin{eqnarray}
Z \left[ J^r_{\mu },K_{\mu }\right] &=&\int \mathcal{D}A_{\mu }\mathcal{D}B_{\mu }\exp\left\{ \,i\,\int
\,d^{n}x\,\left[ \,-\frac{1}{4}\,F_{\mu \nu }^{2}\,+\,{\frac{1}{2}}B_{\mu
}B^{\mu }+\,2\,K_{\mu }\,A^{\mu }\right] \right\} \times  \nonumber \\
&&\times \Pi _{r=1}^{N}\det \left[ i\partial \!\!\!/\,-\,m\,-\,{\frac{1}{%
\sqrt{N}}}(e\,\pa\,+\,g\,\pb)+J\!\!\!/^{\left( r\right) }\,\right] .
\end{eqnarray}
Computing the fermionic determinant up to the quadratic order we have
\begin{eqnarray}
Z \left[ J^r_{\mu },K_{\mu }\right] &=&\int \mathcal{D}A_{\mu }\mathcal{D}B_{\mu }
\exp \frac{\imath}{2}\int \frac{d^{D}k}{(2\pi )^{D}}\left\{ -%
{\tilde{A}}_{\mu }\left[ \theta ^{\mu \nu }k^{2}\left( 1-\lambda \right)
+\lambda \,k^{2}\,g^{\mu \nu }\right] {\tilde{A}}_{\nu }\,+\,{\tilde{B}}%
^{\mu }{\tilde{B}}_{\mu }\right.  \nonumber \\
&&\left. \,+\,\sum_{r=1}^{N}(e\,\frac{\tilde{A}_{\mu }\,}{\sqrt{N}}+\,g\,%
\frac{\tilde{B}_{\mu }}{\sqrt{N}}-\tilde{J}^r_{\mu})\Pi ^{\mu \nu
}(k^2)(e\,\frac{\tilde{A}_{\nu }}{\sqrt{N}}\,+\,g\,\frac{\tilde{B}_{\nu
}}{\sqrt{N}}-\tilde{J}^r_{\nu})+\,2\,{\tilde K}_{\mu }\,{\tilde A}^{\mu }\right\}
\label{zb}\;\;,
\end{eqnarray}

\noindent where $\theta ^{\mu \nu }=g^{\mu \nu }-\frac{k^{\mu }k^{\nu }}{%
k^{2}}$ and the tilde over the fields
represent their Fourier transformations in momentum space.
The quantity $\Pi ^{\mu \nu }$ is the vacuum polarization tensor:

\begin{equation}
\Pi ^{\mu \nu }(k)=i\int \;\frac{d^{D}p}{(2\pi )^{D}}\;tr\left[ \frac{1}{\ps %
-m+i\epsilon }\gamma ^{\mu }\frac{1}{(p\!\!\!/+k\!\!\!/)-m+i\epsilon }\gamma ^{\nu
}\right] \label{pimini}.
\end{equation}

In order to proceed further we have to calculate $\Pi ^{\mu \nu }$ which
depends on the dimensionality of the space-time.

 %\begin{equation}
%L_{eff}^{(2)}[J_{r},K]=-J_{\mu }^{T}\,\Pi ^{\mu \alpha }\left( G_{\alpha
%\beta }\right) ^{-1}\Pi ^{\beta \nu }\,J_{\nu }^{T}-\sum_{r=1}^{N}J_{\mu
%\left( r\right) }\,\Pi ^{\mu \nu }\,J_{\nu \left( r\right) }-\,K_{\mu
%}\,D^{\mu \nu }K_{\nu }+J_{\mu }^{T}\,C^{\mu \nu }\,K_{\nu },
%\end{equation}
%
%\noindent where $J^{T}\equiv \sum_{r=1}^{N}J_{r}^{\nu }$ and
%\[
%G^{\mu \nu }\equiv g^{2}\Pi ^{\mu \nu }+g^{\mu \nu }-\left( \frac{e^{2}}{%
%N\,g^{2}}\right) \left( T^{\mu \nu }\right) ^{-1};
%\]
%\[
%T^{\mu \nu }\equiv g^{\mu \nu }k^{2}-(1-\frac{1}{\lambda })\,k^{\mu }k^{\nu
%}-i\,b\,\in ^{\mu \nu \alpha }k_{\alpha }\,-\left( \frac{e^{2}}{N\,g^{2}}%
%\right) g^{\mu \nu },
%\]
%\[
%D^{\mu \nu }=\,\left( T^{\mu \nu }\right) ^{-1}+\left( \frac{e^{2}}{N\,g^{2}}%
%\right) \left( T^{\mu \alpha }\right) ^{-1}\left( G_{\alpha \beta }\right)
%^{-1}\left( T^{\beta \nu }\right) ^{-1};
%\]
%\begin{equation}
%C^{\mu \nu }=\,\left( \frac{2\,e}{\sqrt{N}\,g}\right) \left( \Pi ^{\mu
%\alpha }\right) ^{-1}\left( G_{\alpha \beta }\right) ^{-1}\left( T^{\beta
%\nu }\right) ^{-1},
%\end{equation}
%
%\noindent with $\lambda $ being the gauge fixing parameter, for a Lorentz
%type one ($L_{GF}\equiv \left( \partial _{\mu }A^{\mu }\right) ^{2}/\left(
%2\,\lambda \right) )$. From the above, one can easily verify that there is a
%contribution which is common for any current-current fermionic correlation
%function, which depends of a collective effect, and for the other hand there
%is also crossed correlation functions which are essentially equal. Now we
%illustrate better by studying the particular case of $D=2$.

\section{Bosonization from two point  correlators in $D=2$}

In this section we restrict ourselves to the $\, D=2 \, $ case. Using dimensional
regularization we obtain,
below the pair creation threshold ($%
z\equiv \frac{k^{2}}{4\,m^{2}}<1$), :

\be
\Pi_{\mu\nu} \, = \, {\tilde \Pi } (k^2) \, \theta_{\mu\nu} \ee

\no with

\begin{equation}
{\tilde{\Pi}}(k^{2})={\frac{1}{\pi }}\left[ 1-\frac{1}{\left[ z\left(
1-z\right) \right] ^{\frac{1}{2}}}\arctan \sqrt{\frac{z}{1-z}}\right]
;\,\,\,\,\,0<\,z\,<\,1,
\end{equation}

\begin{equation}
{\tilde{\Pi}}(k^{2})={\frac{1}{\pi }}\left[ 1-\frac{1}{2}\,\frac{1}{\sqrt{%
z\,(z-1)}}\ln \left( {\frac{\sqrt{(1-{z})}+\sqrt{-z}}{\sqrt{(1-{z})}-\sqrt{-z%
}}}\right) \right] ;\,\,\,\,\,\,\,z<\,0.
\end{equation}

\no Once the tensor $\Pi %
^{\mu \nu }$ is calculated one is left with a gaussian integral over the vector fields
$A_{\mu }$ and $B_{\mu }$ from which a generating functional  quadratic in the sources
is derived. Such generating functional furnishes the following two point
correlators\footnote{More precisely, we should have written explicitly the two point
functions in the form $<G(k)H(p)>= I(k)\delta^{(2)} (k+p)\, $  but in this article we
will not display the delta function for a matter of convenience. Notice also that when
we write $A_{rs}=F + G \delta_{rs}$ it is assumed that $F $ multiplies a $N\times N$
matrix where all entries are equal to one.}

\begin{eqnarray}
<j_{\mu }^{r }\left( k\right) \,j_{\nu }^{s}\left( -k\right)
> &=& \,-\frac{\tilde{\Pi}^{2}\left( e^{2}- k^{2}g^{2}\right) }{N D}
\,\theta ^{\mu \nu }\,+\,\tilde{\Pi}%
\,\theta ^{\mu \nu }\,\delta ^{rs}\, \label{jj} , \\
<j_{\mu }^{\left( r\right) }\left( k\right) \,A_{\nu }\left( -k\right) > &=& \,%
\frac{\,e\,\tilde{\Pi}}{\sqrt{N} D }\,\,\,\theta ^{\mu \nu }, \label{jA} \\
<A^{\mu }\left( k\right) \,A^{\nu }\left( -k\right) > &=& \,\left( -\frac{1 }{\lambda
k^{2}}+\frac{1+g^{2}\tilde{\Pi}}{D }\right) \theta ^{\mu \nu }+\frac{g^{\mu \nu }}{%
\lambda k^{2}}\label{AA},
\end{eqnarray}
\noindent where \be D \, = \, \left[ \tilde{\Pi}\,e^{2}-k^{2}\left(
1+g^{2}\tilde{\Pi}\right) \right]. \ee \no The tensor structure of the above
correlation functions are in full agreement with the corresponding Ward identities
based on the $U\left( 1\right) $ symmetry and it will play a key role in our
bosonization procedure as in \cite{bm}.

%\[
%L_{eff}^{(2)}[J,K]=-J_{\mu }\frac{N\,\,\tilde{\Pi}\,a\,k^{2}}{\left[ \tilde{%
%\Pi}\,e^{2}-a\,k^{2}\left( 1+g^{2}\tilde{\Pi}\right) \right] }\,\,\theta
%^{\mu \nu }J_{\nu }-K_{\mu }\frac{1+g^{2}\tilde{\Pi}}{\tilde{\Pi}%
%\,e^{2}-a\,k^{2}\left( 1+g^{2}\tilde{\Pi}\right) }K_{\nu }+
%\]
%\begin{equation}
%+K_{\mu }\frac{2\,\sqrt{N}\,e\,\tilde{\Pi}}{\left[ \tilde{\Pi}%
%\,e^{2}-a\,k^{2}\left( 1+g^{2}\tilde{\Pi}\right) \right] }\,\,\,\theta ^{\mu
%\nu }J_{\nu }.
%\end{equation}

Now, in order to derive a bosonized expression for the currents $j_{\mu }^r =
\overline{\psi }_r\,\gamma_{\mu }\psi_r $, we write down the most general
decomposition for a vector in the momentum space:
\begin{equation}
j_{\beta }^r(k) \, = \, \,\epsilon_{\beta \delta }k^{\delta }\,\phi^r(k) \,+\,k_{\beta
}\,\varphi^r (k).
\end{equation}

\noindent Substituting it in (\ref{jj}) and using the identity $\epsilon _{\beta
\delta }k^{\delta }\epsilon_{\alpha \gamma }k^{\gamma }= k^{2}\theta _{\alpha \beta }$
we obtain
\begin{eqnarray}
<j^r_{\alpha }\left( k\right) \,j^s_{\beta }\left( -k\right) >&=& -k^{2}\theta
_{\alpha \beta }\,<\phi^r\left( k\right) \phi^s \left( -k\right) >-k_{\alpha }k_{\beta
}<\varphi^r \left( k\right) \varphi^s \left( -k\right) > \nn \\
&-&\epsilon _{\alpha \delta }k^{\delta }k_{\beta }<\phi^r\left( k\right) \varphi^s
\left( -k\right) >-\epsilon _{\beta \gamma }k^{\gamma }k_{\alpha }<\varphi^r \left(
k\right) \phi^s\left( -k\right) > \nn \\
&=& -\frac{\tilde{\Pi}^{2}\left( e^{2}- k^{2}g^{2}\right) }{N D}
\,\theta ^{\mu \nu }\,+\,\tilde{\Pi}%
\,\theta ^{\mu \nu }\,\delta ^{rs}.
\end{eqnarray}

\noindent From the above it is not difficult to derive
\begin{eqnarray}
<\varphi^r \left( k\right) \varphi^s \left( -k\right) >&=& 0 , \label{vfvf}\\
<\phi^r \left( k\right) \varphi^s \left( -k\right) > &=& 0 , \label{fvf} \\
<\phi^r \left( k\right) \phi^s \left( -k\right) > &=& -\frac 1{k^2}\left\lbrack
\,-\frac{\tilde{\Pi}^{2}\left( e^{2}- k^{2}g^{2}\right) }{N D}
\,+\,\tilde{\Pi}%
\,\,\delta ^{rs}\right\rbrack , \label{ff} \end{eqnarray}

\noindent from which one can safely set $\varphi =0$. On the other hand, substituting
a general decomposition
\begin{equation}
<\phi \left( k\right) A_{\mu }\left( -k\right) >\,=\,M\,\in _{\mu \delta }k^{\delta
}\,+\,Q\,k_{\mu },
\end{equation}
in the mixed correlation functions $ <j^r_{\mu }( k) A_{\nu }( -k )
> $ given in (\ref{jA}) we conclude that
\begin{equation}
<\phi^r \left( k\right) A_{\mu }\left( -k\right) >\, =\, -\frac{e
\tilde{\Pi}}{\sqrt{N}k^2 D}\,\epsilon_{\mu \delta }k^{\delta };\,
\end{equation}

%\noindent and
%\begin{equation}
%<A_{\mu }\left( k\right) A_{\nu }\left( -k\right) >=\frac{g^{2}\left(
%1+\lambda \right) }{e^{2}}\,g_{\mu \nu }-\left[ \frac{g^{2}}{e^{2}}\left(
%1+\lambda \right) +\frac{1+g^{2}\tilde{\Pi}}{D}\right] \theta _{\mu \nu }.
%\end{equation}

Now we are in a position to derive the bosonic Lagrangian density $\, {\cal L}_B
(A_{\mu},\phi^r) \, $ which is compatible with the correlation functions
(\ref{jj}),(\ref{jA}) and (\ref{AA}). For this purpose we start from the following
Ansatz
\begin{equation}
{\cal L}_{B}(A_{\mu},\phi^r) \,=\,\phi^r \,R_{rs}\,\phi^s +2\,S_r\,\phi^r \,\epsilon
_{\mu \nu }k^{\nu }A^{\mu }+A^{\mu }A^{\nu }\left( T_{1}\,\theta _{\mu \nu
}\,+\,T_{2}\,g_{\mu \nu }\right),
\end{equation}
\no where $R_{rs}\, , \, S_r \, , \, T_1 $ and $\, T_2$ will be determined as follows.
We introduce the external sources $X^r $ and $  K_{\mu} $ and define the generating
functional

\be Z_B\left[ X_r,K_{\mu }\right] \, = \, \int \prod_{r=1}^N{\cal D}\phi^r {\cal
D}A_{\mu } e^{i \int d^2x\left[{\cal L}_B(A_{\mu},\phi^r)+ X_r \phi^r + K_{\mu
}A^{\mu}\right]} \label{star}. \ee

\no Assuming that  $R_{rs}$ is a symmetric non-singular matrix we have performed the
gaussian integrals in (\ref{star}) and obtained an explicit formula for $Z_B\left[
X^r,K_{\mu }\right]$ from which the two-point correlators can be obtained. By matching
those correlators with (\ref{jj}),(\ref{jA}) and (\ref{AA}) we determine the bosonic
Lagrangian uniquely :

\bea {\cal L}_B(A_{\mu},\phi^r)  \, &=& \, \frac 12\left[ \sum_{r=1}^N \phi^r
\frac {k^2}{{\tilde
\Pi}}\phi^r + \frac{g^2`}{N} \, k^2 \left(\sum_{r=1}^N \phi^r
\right)\left(\sum_{s=1}^N \phi^s\right)\right]  \nn \\
& &  - \frac{A^{\alpha}A^{\beta}}2\left[(1-\lambda )k^2\theta_{\alpha\beta } + \lambda
k^2 g_{\alpha\beta} \right] \nn \\
& & + \frac e{\sqrt{N}} \, \epsilon_{\mu\nu} A^{\mu}k^{\nu} \left( \sum_{r=1}^N \phi^r
\right)\label{A}\quad. \eea

\no If we quantize ${\cal L}_B(A_{\mu},\phi^r) $ and integrate over the scalar fields
$\phi^r $ in (\ref{A}) this will lead to a nonlocal effective action for the photon
which was studied in \cite{ddh2} where we concluded that, although non local, the
theory is free of tachyons. Next, by comparing with the Lagrangian density written in
terms of fermionic fields in (1)  we have the bosonization formulae for each fermion
flavor (no sum over repeated indices below) :

\bea \overline{\psi }_r\,\gamma_{\mu }\psi_r (k)\, &=& \, \epsilon_{\mu\nu} k^{\nu}
\phi^r (k), \label{A1} \\
-\bar{\psi}_{r}(k)\,(k \!\!\!/\,+\,m\, )\psi_{r} (k)\, &=& \, \frac 12 \phi^r \frac
{k^2}{{\tilde \Pi}}\phi^r . \label{A2} \eea

\no Now some comments are in order. First of all, if for some given flavor we do a
$U(1)$ transformation ($\psi_r\to e^{i\alpha } \psi_r $ )  in the expectation value
$<j_{\mu}^r >$ and use any regularization scheme preserving the $U(1)$ symmetry it
will be easy to derive the Ward identity $<j^r_{\mu}\partial^{\nu}j^r_{\nu}> =0 $
which implies the tensor structure $<j^r_{\mu}j^r_{\nu}>\, \propto \,\theta_{\mu\nu} $
and consequently we will have (\ref{A1}). So the current is topological due to the
$U(1)$ global symmetry and that must hold non-perturbatively. On the other hand, the
bosonization rule (\ref{A2}) is only approximate since the full expression would
require, in our approach, the complete knowledge of the fermionic determinant which is
only possible for $m\to 0$. In this case $\tilde{\Pi }\to 1/\pi $ and we end up with
$N$ massless scalar fields topologically coupled to the gauge field. Integration over
the gauge field leads to $N-1$ massless scalar modes and one mode with $\, m^2 =
e^2/(N\pi + g^2) \, $. Thus, reproducing the particular case of the so called
Schwinger-Thirring model for $N=1$ \cite{st}, as well as the Schwinger model result
$m^2=e^2/\pi $ for $g\to 0 $ and $N=1$.

\section{Bosonization from two-point correlators in $D=3$}


In $\, D=3 \, $ dimensions the vacuum polarization tensor (\ref{pimini}) calculated by
means of dimensional regularization is given by :

\begin{equation}
\Pi ^{\mu \nu }(k)=i\,\,\,\Pi _{1}\,E^{\mu \nu }+\Pi _{2}k^{2}\,\theta ^{\mu \nu
}\label{pimini3} \, ,
\end{equation}

\noindent with $E^{\mu \nu }\equiv \epsilon ^{\mu \nu \rho }k_{\rho }$ and, in the
range $0\leq \,z\,<\,1$,

\begin{equation}
\Pi _{1} \,=\,-\frac{1}{8\pi z^{1/2}}\ln \left( \frac{%
1+z^{1/2}}{1-z^{1/2}}\right) ;\,\qquad \Pi _{2}=\,\frac{1}{16\pi m\, z}\left[ 1-\left(
\frac{1+z}{2 z^{1/2}}\right)\ln \left( \frac{%
1+z^{1/2}}{1-z^{1/2}}\right)  \right] .
\end{equation}
While for  $\, z<0 \, $ we have

\begin{equation}
\Pi _{1} \,=\, -\frac{1}{4\pi (-z)^{1/2}}\arctan \sqrt{-z};\, \,\, \,\, \Pi
_{2}=\,\frac{1}{16\pi m\, z}\left[ 1-\left( \frac{1+z}{2 (-z)^{1/2}}\right)\arctan
\sqrt{-z}\right] . \label{p1p2b}
\end{equation}

\no Substituting (\ref{pimini3}) in the general expression (\ref{zb}) we can obtain
the two-point functions:

\begin{eqnarray}
<j_{\mu }^{r }( k) \,j_{\nu }^{s}(-k)
> &=& -\frac 1N \left[k^2\left(\Pi_2 +\frac P{\tilde Q} \right)
\,\theta_{\mu \nu }\,+\, i\,\Pi_1\left(1-\frac{k^2}{\tilde Q}\right) E_{\mu\nu} \right] \nn \\
&+& \left( k^2\Pi_2 \theta_{\mu\nu} + i\,\Pi_1 E_{\mu\nu} \right) \delta_{rs} , \label{jj3}\\
<j_{\mu }^{r}\left( k\right) \,A_{\nu }\left( -k\right) > &=& \,%
\frac{e}{\tilde{Q}\sqrt{N} }\,
\left[P\theta _{\mu \nu }- i\,\Pi_1 E_{\mu\nu} \right] , \label{jA3} \\
<A_{\mu }\left( k\right) \,A_{\nu }\left( -k\right) > &=& \frac{g_{\mu\nu}}{\lambda
k^{2}}+ \left[\frac{e^2P}{k^2\tilde{Q}}-\frac{(1+\lambda )}{\lambda k^2}
\right]\theta_{\nu\mu} -\frac{i e^2\Pi_1}{k^2\tilde{Q}}E_{\mu\nu} , \label{AA3}
\end{eqnarray}


\no where we found convenient to define

\bea
P\, &=& \, (e^2-k^2 g^2)(k^2\Pi_2^2-\Pi_1^2)-k^2 \Pi_2 \quad , \label{P} \\
\tilde{Q} \, &=& \, k^2\left[(e^2-k^2 g^2)\Pi_2 -1\right]^2 - (e^2-k^2 g^2)^2\Pi_1^2
\quad . \label{tildeQ} \eea

\noindent In Analogy to the $D=2$ case, we use for the $U(1)$ current a general
decomposition in the momentum space :

\begin{equation}
j^r_{\alpha }(k) \, = \,\epsilon _{\alpha \beta \gamma }k^{\beta }\,B_r^{\gamma
}(k)\,+\,k_{\alpha }\,\phi^r (k), \label{d3} \end{equation}

\noindent from which we get

\bea <j^r_{\alpha }\left( k\right) \,j^s_{\beta }\left( -k\right) >&=&-k_{\alpha
}k_{\beta }<\phi^r \left( k\right) \phi^s \left( -k\right)
>+\,E_{\alpha \gamma }E_{\delta \beta
}<B_r^{\gamma }\left( k\right) B_s^{\delta }\left( -k\right) > \nn \\
&+&\,k_{\alpha }\,E_{\beta \delta }\,<\phi^r \left( k\right) B_s^{\delta }\left(
-k\right) >+\,k_{\beta }\,E_{\alpha \gamma }<B_r^{\gamma }\left( k\right) \phi_s
\left( -k\right) >. \eea

\no Multiplying the last expression by $k^{\alpha}k^{\beta}$ we conclude that
$<\phi^r\left( k\right)\phi^s \left(- k\right)> = 0 $. Now, multiplying the resulting
expression by $k^{\alpha }$ we have $E_{\alpha\beta }<\phi^r(k) B_s^{\beta}(-k)>=0$. A
similar manipulation was used in the last section to derive (\ref{vfvf}),(\ref{fvf})
and (\ref{ff}). Concluding, we can certainly neglect the scalar fields $\phi^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 identically
conserved. As in $D=2$ case this happens because of the $U(1)$ global symmetry of the
fermionic Lagrangian. Taking $\phi^r=0$ and substituting the decomposition (\ref{d3})
in (\ref{jj3}) and (\ref{jA3}), after some trivial manipulations, we end up with \bea
<B_r^{\gamma}(k)B_s^{\delta}(-k)> \, &=& \, -C_{rs}g^{\gamma\delta} +
\left(C-\frac{H_2}{k^2}\right)_{rs}\theta^{\gamma\delta}-\frac{(H_1)_{rs}}{k^2}E^{\gamma\delta}
, \label{BB}\\
<A^{\nu}(k)B_s^{\mu}(-k)> \, &=& \,
-\left(\frac{ie\Pi_1}{\sqrt{N}\tilde{Q}}+D\right)_s g^{\nu\mu} - \left(\frac{e
P}{\sqrt{N}\tilde{Q}k^2}\right)_s E^{\nu\mu}+ D_s \theta^{\nu\mu} , \label{AB} \eea

\no where $C_{rs}$ and $D_s $ are arbitrary and

\bea (H_1)_{rs} \, &=& \, i\,\Pi_1 \,\delta_{rs} + \frac{i\,\Pi_1}N
\left(\frac{k^2}{\tilde{Q}}-1
\right) , \label{H1} \\
(H_2)_{rs} \, &=& \, k^2\Pi_2 \,\delta_{rs} - \frac{k^2}N\left( \Pi_2 +\frac
P{\tilde{Q}} \right). \label{H2} \eea

\no The arbitrariness of $ C_{rs} $ and $D_s$ shows that the bosonization fields
$B_r^{\mu}$ must be gauge fields but the corresponding gauge symmetry is independent
of the eletromagnetic one. Analogous to the last section, we next derive a Lagrangian
density compatible with the two-point correlators (\ref{BB}),(\ref{AB}) and
(\ref{AA3}). Let us suppose we have a bosonic Lagrangian density ${\cal
L}_B(A_{\mu},B^{\nu}_r) $ of the form

\bea {\cal L}_{B}(A_{\mu},B^{\nu}_r)\, &=& \,B_r^{\mu }\,a^{rs}\, \theta_{\mu \nu }\,B_s^{\nu
}+ B_r^{\mu }\,b^{rs}\, E_{\mu \nu }\,B_s^{\nu }+ B_r^{\mu }\,d^{rs}\, g_{\mu \nu
}\,B_s^{\nu }\nn\\
&+& u^s B^{\mu }\,E_{\mu \nu }\,\,A^{\nu }+A_{\mu }A_{\nu }\left(v_1 \,
\theta_{\mu\nu} + v_2 g_{\mu\nu} \right) , \label{lb3} \eea

\noindent with $a^{rs},b^{rs},d^{rs},u^s, v_1, v_2 $ to be determined by matching the
two-point correlators  derived from the generating functional :


\be Z_B\left[ Y_{\nu}^r,K_{\mu }\right] \, = \, \int \prod_{r=1}^N{\cal D}B^{\mu}_r
{\cal D}A_{\mu } e^{i \int d^2x\left({\cal L}_{B}(A_{\mu},B^{\nu}_r) + Y^r_{\mu}
B_r^{\mu } + K_{\mu} A^{\mu}\right)}, \label{star3}\ee

\no with the correlators (\ref{BB}),(\ref{AB}) and (\ref{AA3}). Assuming that the
matrices $a^{rs},b^{rs},d^{rs}$ are symmetric and non-singular, after a long but
straightforward calculation, we obtain the bosonic Lagrangian density in the momentum
space:

\bea {\cal L}_B(A_{\mu},B_r^{\alpha})  &=&
-\frac 12\left[ \sum_{r=1}^N  \frac {B^{\alpha}_r\left(k^2 \Pi_2\,
\theta_{\alpha\beta} + i\, \Pi_1 \epsilon_{\alpha\beta\gamma}k^{\gamma}\right)B^{\beta}_r }
{k^2 \Pi_2^2 - \Pi_1^2}+ \frac{g^2}{N} \, \left(\sum_{r=1}^N
B^{\alpha}_r
\right)k^2\, \theta_{\alpha\beta}\left(\sum_{s=1}^N B^{\beta}_s\right)\right]  \nn \\
& &  - \frac{A^{\alpha}A^{\beta}}2\left[(1-\lambda )k^2\theta_{\alpha\beta } + \lambda
k^2 g_{\alpha\beta} \right]+ \frac e{\sqrt{N}} \,
 \epsilon_{\mu\nu\gamma} k^{\gamma}\, A^{\mu}  \sum_{r=1}^N B^{\nu}_r
\nn \\
& & + \sum_{r=1}^N\frac{B_r^{\alpha}B^{\beta}_r}2\left[\tilde{\lambda}
k^2\theta_{\alpha\beta } - \tilde{\lambda} k^2 g_{\alpha\beta} \right] \label{A3}\quad
. \eea

\no Notice that $\lambda $ and $\tilde{\lambda}$ are independent gauge fixing
parameters. Integration over the vector fields $B^{\alpha}_r$ will lead
to an effective eletromagnetic theory which was also
studied in \cite{ddh2} ( see also \cite{ddh1} ) where we have
analysed its pole structure and concluded that no
tachyons appear just like in its two dimensional couterpart (\ref{A}).

Comparing (\ref{A3}) with the original Lagrangian
(1) written in terms of fermionic fields
we have the bosonization rules for each fermion flavor ($\, r=1,2,\cdots , N \, $):

\bea \overline{\psi }_r\,\gamma_{\mu }\psi_r (k)\, &=& \, \epsilon_{\mu\nu\alpha} k^{\nu}
B^{\alpha}_r(k) , \label{A13} \\
-\bar{\psi}_{r}(k)\,(k \!\!\!/\,+\,m\, )\psi_{r} (k)\, &=& \, -\frac 12 B^{\alpha}_r
\frac {\left(k^2 \Pi_2\, \theta_{\alpha\beta} + i\, \Pi_1 E_{\alpha\beta}\right)} {k^2
\Pi_2^2 - \Pi_1^2}B^{\beta}_r . \label{A23} \eea

Similarly to the $D=2$ case, we have obtained (\ref{A13}) from the fact that for each
fermion flavor we have the tensor structure $<j_{\mu}j_{\nu}> \, \propto a\,
\theta_{\mu\nu} + b \, E_{\mu\nu} $, which has been derived by calculating the
fermionic determinant up to the quadratic order. However, this tensor structure must
hold beyond that approximation ( non-perturbatively ) as a consequence of the Ward
identity $<j_{\mu}\partial^{\nu}j_{\nu}>=0\, $ which must be true also in $D=3$ by
using a gauge invariant regularization. On the other hand, the bosonization rule
(\ref{A23}) becomes exact only in the limit $m\to 0 $. Taking $\, m\to 0 \, $ (or $\,
z\to -\infty\, $) in the expressions (\ref{p1p2b}) we have $\Pi_1\to 0 $ and $\Pi_2\to
1/(16\sqrt{-k^2})$. Plugging back in (\ref{A13}) and (\ref{A23}) we recover the result
of \cite{marino} with $\beta=1/4$ and $\theta=0$ in that reference (see also
\cite{bfo} for finite fermion mass). We stress however, that in both \cite{marino} and
\cite{bfo} only free fermions have been considered while our results have been derived
for an interacting theory with Thirring and eletromagnetic couplings.

\section{Conclusions}

We have derived bosonization maps
 for the $U(1)$ currents and the fermion Lagrangian densities for both
$QED_{2}$ and $QED_{3}$ with $N$ fermion flavors and Thirring self-interaction. Both
results hold for finite fermion masses and no derivative expansion on $\frac km $ is
made as in \cite{sghosh}. By turning off the interactions we can reproduce the results
of \cite{bfo} 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 (topological) as a direct consequence of the
Ward-identity $\, <j_{\mu}\partial^{\nu}j_{\nu}>\, =\, 0  $ which must be true
non-perturbatively if there is no $U(1)$ anomaly. In particular, there is no need for
looking at higher point functions  or compute the fermionic determinant beyond the
quadratic approximation to confirm that. On the other hand, our maps for the
Lagrangian densities (\ref{A2}) and (\ref{A23}), which are also independent of the
interactions, hold  only perturbatively  due to our quadratic approximation for the
fermionic determinant which amounts to neglect terms of order $1/N \,$ and quartic in
the couplings $g$ and $e$ and terms of order $1/N^2$  and higher in the two point
current correlators as well as higher point current correlators.

Finally, we should mention that the results derived here could have been obtained by a
technically more direct way along the lines of \cite{bos} (see also \cite{bq}) but our
calculations do not use any auxiliary field and clarifies the fundamental role of the
two point correlators. Besides, it might be also useful ( work in progress ) for
deriving approximate bosonic maps for other fermion bilinears like the mass term $\,
{\overline\psi}\psi\, $ in $D=3$, which apparently can not be done in the approach of
\cite{bos}.

\section{Acknowledgements}

This work was partially supported by \textbf{CNPq} and \textbf{FAPESP},
Brazilian research agencies.

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\newpage

\end{document}

