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\begin{document}
\title{Low-Energy-Theorem Approach to one-particle singularity in QED$_{2+1}$}
\author{Yuichi Hoshino}
\affiliation{Kushiro National College of Technology,Otanoshike nishi 2-32-1,Kushiro,
Hokkaido 084,Japan}

\begin{abstract}
We evaluate the spectral function of scalar and spinor propagator in three
dimensional quantum electrodynamics with the use of Ward-Identity for
soft-photon emission vertex.We work well in position space to treat infrared
divergences in our model. Exponentiation of one-photon matrix element yields a
full propagator in position space.It has asimple form as free propagator
multiplied by quantum correction.But this is not an integrable function so
that analysis in momentum space is not easy.Term by term integral converges
and they have a logarithmic singularity associated with renormalized mass in
perturbation theory$.$Renormalization constant vanishes for weak
coupling,which suggests confinement of charged particle.There exsists a
critical coupling constant below which bare mass vanishes and the vacuum
expectation value of pair condensation is finite.

\end{abstract}
\maketitle
\tableofcontents


\section{ Introduction}

In three dimensional gauge theory infrared divergences is severer than that in
four dimension.In 1981,Jackiw first demonstrated it in the massless fermionic
$QED_{2+1}$ and after that other authors introduced the infrared counter terms
to renormalize the infrared divergences[1,2].There had been an attempts to
solve Dyson-Shwnger equations to examine the dynamical mass generation or
chiral symmetry breakings in this model to improve low-energy
behaviour[3,4].Another important feature of this model is that it allows
parity violating Chern-Simons term in the Lagrangean without violating gauge
invariance[5].But it has not yet been clear the quantum effects of
Chern-Simons term.During the same period infrared behaviour of the propagator
in the presence of dynamical mass has been analysed to search the physical cut
associated with massless photon[6,7].To maintain gauge covariance Delbourgo
and Jackiw addded the vertex correction with Ward-Identities for soft
particles in the determination of the infrared behaviour of the propagator
with bare mass[9,10].Their works are fundamental and important to understand
the effects of soft-photons in comparison with perturbation theory.Of course
their works includes non-perurbative effects.In 1992 Delbourgo applied his
method(gauge thecnique) to $QED_{2+1}$ and shown that massless loop correction
to photon soften infrared divergences as T.Appelquist et.al[3,6].In
reference[9] infrared behaviour of the scalar propagator in the presence of
massless fields (photon, graviton) in four dimension was determined by solving
the spectral function.In this work we apply the same method in three dimension
to determine the infrared behaviour of the propagator.In section I we analyse
scalar QED$_{3}$ and evaluate the scalar propagator in position space.In this
case an approximaiton is made to choose the one-photon,one-meson intermediate
state to derive the matrix element $\left\langle \Omega|\phi(x)|n\right\rangle
$ for the spectral function.Most singular infrared part is assumed to be the
contribution of soft photon emitted from external lines.Therefore we introduce
photon mass to avoid infrared divergences.Lowest order spectral function
contains mass and wave function renormalization up to logarthum in position
space. Exponentiation of the lowest-order spectral function yields the full
propagator in position space. Especially mass renormalization make a drastic
change of the propagator.Its has a simple form as free propagator (with
renormalized mass) multiplied by quantum correction.However this function is
not integrable and it is difficult to make a fourier transform of it.In the
pertuebative analysises the propagator have linear and logarithmic infrared
divergences near on-shell.We mention the gauge transformation property and see
our solution satisfies Landau-Kharatonikov transformation.As far as the
renormalization constant is concerned we evaluate it by position space
propagator.The result is $Z=0$ for weak coupling,which is a signpost of
confining phase in our approximation.In section II we study QED$_{3}$ with
spinor and see the spectral function for fermion is the same in section I.In
this case there is a interesting possibility of pair condensation.Evaluating
the vacuume expectation value $\left\langle \overline{\psi}\psi\right\rangle $
and we find that there exists a critical coupling constant ($e^{2}/(8\pi
m)=1)$ below which the bare mass vanishes and $\left\langle \overline{\psi
}\psi\right\rangle $ remains finite.

\section{ Scalar QED}

First we assume the Kallen-Lehmann spectral representation of the propagator
for massive boson[1]:%

\begin{align}
\Delta(p^{2})  &  =\int d^{3}x\exp(ip\cdot x)\left\langle \Omega|T\phi
(x)\phi^{+}(0)|\Omega\right\rangle ,\\
\Delta(p^{2})  &  =\int\frac{\sigma(s)ds}{p^{2}-s+i\epsilon},\\
\sigma(p^{2})  &  =(2\pi)^{2}\sum_{N}\delta(p-p_{N})\left\langle \Omega
|\phi|N\right\rangle \left\langle N|\phi^{+}|\Omega\right\rangle ,\nonumber\\
|N  &  >=|r;k_{1},.....k_{n}>,r^{2}=m^{2}.
\end{align}
The spectral function is formally written as a sum of multi-photon
intermideate states:%
\begin{align}
\sigma(p^{2})  &  =\int\frac{d^{2}r}{2r_{0}}\sum_{n=0}^{\infty}\frac{1}%
{n!}(\int\frac{d^{3}k}{(2\pi)^{2}}\theta(k_{0})\delta^{(3)}(k^{2}%
)\sum_{\epsilon})_{n}\delta(p-r-\sum_{i=1}^{n}k_{i})\nonumber\\
&  \times\left\langle \Omega|\phi|r_{i};k_{1},...k_{n}\right\rangle
\left\langle r;k_{1},..k_{n}|\phi^{+}|\Omega\right\rangle .
\end{align}
Here we use the {\normalsize notation}%
\begin{align}
(f(k))_{0}  &  =1,\nonumber\\
(f(k))_{n}  &  =\prod_{i=1}^{n}f(k_{i}).
\end{align}
To evaluate the contribution of the soft-photons,we consider from the
beginnings when only the $n$th photon is soft.Define the following matrix
element%
\begin{equation}
T_{n}(r;k_{1},..,k_{n})=\left\langle \Omega|\phi|r;k_{1},..,k_{n}\right\rangle
.
\end{equation}
We consider $T_{n}$ for $k_{n}^{2}\neq0,$we continue off the photon mass-shell
by Lehmann-Symanzik-Zimmermann (LSZ) formula:%

\begin{align}
T_{n}  &  =\epsilon_{n}^{\mu}T_{n\mu},\\
T_{n}^{\mu}  &  =\int d^{3}x\exp(ik_{n}\cdot x)\square_{x}\left\langle
\Omega|T\phi A_{\mu}(x)|r;k_{1},..,k_{n}\right\rangle \nonumber\\
&  =\int d^{3}x\exp(ik_{n}\cdot x)\left\langle \Omega|T\phi j^{\mu}%
(x)|r;k_{1},...,k_{n-1}\right\rangle ,
\end{align}
where the electromagnetic current is%
\begin{equation}
j^{\mu}(x)=ie\phi^{+}(x)\overleftrightarrow{\partial_{\mu}}\phi(x)-2e^{2}%
A_{\mu}\phi^{+}\phi,
\end{equation}
here we assume the second term does not contribute in the infared behaviour of
the propagator and the usual commutation relation for the first term%
\begin{equation}
\delta(x_{0}-y_{0})[j_{0}(x),\phi(y)]=e\phi(x).
\end{equation}
Thus we omitt the second term in the present calculus.From the definition
(8),$T_{n}^{\mu}$ is seen to satisfy the Ward-Identity%
\begin{equation}
k_{n\mu}T_{n}^{\mu}=eT_{n-1}(r;k_{1},..,k_{n-1}).
\end{equation}
Here we mention the simple proof:%
\begin{align}
k_{n\mu}T_{n}^{\mu}  &  =\int d^{3}xk_{n\mu}\epsilon_{n}^{\mu}\exp(ik_{n}\cdot
x)\left\langle \Omega|T\phi j^{\mu}(x)|r;k_{1},...,k_{n-1}\right\rangle
\nonumber\\
&  =i\int d^{3}x\epsilon_{n}^{\mu}\exp(ik_{n}\cdot x)\partial_{\mu
}\left\langle \Omega|T\phi j^{\mu}(x)|r;k_{1},...,k_{n-1}\right\rangle
\nonumber\\
&  =e\epsilon_{n}^{\mu}\left\langle \Omega|\phi|r;k_{1},..,k_{n-1}%
\right\rangle ,
\end{align}
provided%
\begin{equation}
\partial_{\mu}T(\phi(y)j^{\mu}(x))=\delta(x-y)e\phi(y),
\end{equation}
\qquad\qquad where $e$ is the charge carried by meson.To get the solution of
the Ward-Identity we seek the non-singular contribution of photon for
$T_{n}(r;k_{1},..,k_{n}).$There is a possible solution which satisfy
Ward-Identity[9]%
\begin{equation}
T_{n}^{\mu}(r;k_{1}..k_{n})=\frac{e(2r+k_{n})^{\mu}}{2r\cdot k_{n}+k_{n}^{2}%
}T_{n-1}(r,k_{1}...k_{n-1}),r^{2}=m^{2}.
\end{equation}


Here we assume most singular contributions of photons emitted from external
lines as in four dimension.These arise from Feynman diagram in which the
soft-photon line with momentum $k_{n}^{\mu}$ and the incoming meson line can
be separated from the remainder of the diagram by cutting a single meson
line.Detailed dicussion and evaluation of the explicit form of $T_{n}$ are
given in ref [9].Of course for $n=1$%
\begin{equation}
T_{1}=\frac{e(2r+k)\cdot\epsilon}{(2r\cdot k+k^{2})}(x=0),
\end{equation}
and we can determine $T_{n}$ recursively with eq(15).In this case $T_{n}$
becomes
\begin{equation}
T_{n}=\left(  \frac{e(2r+k)\cdot\epsilon}{2r\cdot k+k^{2}}\right)  _{n}.
\end{equation}
The one photon matrix element by LSZ is%
\begin{align}
T_{1}  &  =\left\langle in|T(\phi_{in}(x),ie\int d^{3}y\phi_{in}%
^{+}\overleftrightarrow{\partial_{\mu}^{y}}\phi_{in}(y)A_{in}^{\mu
}(y))|r;k\text{ }in\right\rangle \nonumber\\
&  =ie\int d^{3}yd^{3}z\triangle_{F}(x-y)\overleftrightarrow{\partial_{\mu
}^{y}}\epsilon_{\mu}^{\lambda}(k)\exp(ik\cdot y)\delta^{(3)}(y-z)\exp(ir\cdot
z)\nonumber\\
&  =e\frac{(2r+k)_{\mu}\epsilon_{\mu}^{\lambda}(k)}{(r+k)^{2}-m^{2}}%
\exp(i(r+k)\cdot x).
\end{align}
From this lowest order matrix element the function in (4) is given%
\begin{align}
F  &  =\sum_{one\text{ }photon}\left\langle \Omega|\phi|r;k\right\rangle
\left\langle r;k|\phi^{+}|\Omega\right\rangle \nonumber\\
&  =\int\frac{d^{3}k}{(2\pi)^{2}}\exp(ik\cdot x)\delta(k^{2})\theta
(k^{0})[\frac{e^{2}(2r+k)^{\mu}(2r+k)^{\nu}\Pi_{\mu\nu}}{(2r\cdot k+k^{2}%
)^{2}}],
\end{align}
and $\sigma$ is expressed by
\begin{equation}
\sigma(p)=\int\frac{d^{3}x}{(2\pi)^{3}}\exp(ip\cdot x)\int\frac{d^{2}r}%
{2r^{0}}\exp(ir\cdot x)\exp(F).
\end{equation}
Here $\Pi_{\mu\nu}$ is a polarization sum we have%
\begin{equation}
\Pi_{\mu\nu}=-(g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{k^{2}})-d\frac{k_{\mu}k_{\nu}%
}{k^{2}},
\end{equation}
and the free photon propagator%
\begin{equation}
D_{0}^{\mu\nu}=\frac{1}{k^{2}+i\epsilon}[g^{\mu\nu}-\frac{k^{\mu}k^{\nu}%
}{k^{2}}+d\frac{k^{\mu}k^{\nu}}{k^{2}}].
\end{equation}
we get%
\begin{align}
F  &  =-e^{2}\int\frac{d^{3}k}{(2\pi)^{2}}\exp(ik\cdot x)\theta(k^{0}%
)\nonumber\\
&  \times\lbrack\frac{(2r+k)^{2}\delta(k^{2})}{(2r\cdot k+k^{2})^{2}%
}+(d-1)\frac{\delta(k^{2})}{k^{2}}].
\end{align}
It is natural to set $\delta(k^{2})/k^{2}$ equals to $-\delta^{^{\prime}%
}(k^{2}).$We see the reason in calculating the one loop self energy of meson
diagram,and take imaginary part.This expression is the same as the one given
above with the interpretation $\delta(k^{2})/k^{2}=-\delta^{^{\prime}}%
(k^{2}).$We must introduce a small photon mass $\mu$ as an infrared cut
off.Therefore ( 22\ ) becomes
\begin{equation}
F=-e^{2}\int\frac{d^{3}k}{(2\pi)^{2}}\theta(k^{0})\exp(ik\cdot x)[\delta
(k^{2}-\mu^{2})(\frac{m^{2}}{(r\cdot k)^{2}}+\frac{1}{(r\cdot k)}%
)-(d-1)\frac{\partial}{\partial k^{2}}\delta(k^{2}-\mu^{2})].{}%
\end{equation}
The evaluation of the integral to $\sigma$ is described in Appendix .The
result is%
\begin{equation}
F=\frac{e^{2}}{8\pi\mu}(d-2)+\frac{\gamma e^{2}}{8\pi r}+\frac{e^{2}}{8\pi
r}\ln(\mu x)-\frac{e^{2}}{8\pi}x\ln(\mu x)-\frac{e^{2}}{8\pi}x(d+\gamma-2).
\end{equation}
It is helpful to use position space to evaluate $F$ that it shows us easily
the short and long-distance bahaviour. If we put $x=0,$usaual perturbation see
infrared and ultraviolet divergences and the answer is reguralization
dependent. Here we think about the physical meanings of each term.
\begin{align}
\sigma(x)  &  =\frac{\exp(-mx)}{4\pi x}\exp(F(x,e,\mu,m)),\\
\sigma(p)  &  =F.T.(\sigma(x))=\int\frac{d^{3}x}{(2\pi)^{3}}\exp(ip\cdot
x)\sigma(x),
\end{align}


The terms proportional to $x$ correspond to mass renormalization and others to
wave function renormalization.
\begin{align}
\sigma(x)  &  =\exp(A)\frac{\exp(-(m+B)x)}{4\pi x}(\mu x)^{-Cx+D},\\
A  &  =\frac{e^{2}}{8\pi\mu}(d-2)+\frac{\gamma e^{2}}{8\pi m},B=\frac{e^{2}%
}{8\pi}(d+\gamma-2),C=\frac{e^{2}}{8\pi},D=\frac{e^{2}}{8\pi m}.
\end{align}
From the form of $\sigma(x)$,wave function renormalization effects denoted by
constant and $D\ln(\mu x)$ modifies the dimension of the propagator as
anomalous dimension in four dimension.Mass renormalization is proportional to
$x$ and $x\ln(\mu x).$Logarithmic corrections descrive the logarithmic
infrared divergences and adds a drastic effect as a factor $(\mu x)^{-Cx}$.In
perturbative annalysis these will be shown clear. By the definition
(1),(2),(3) $\sigma(x)$ is a full puropagator.Free propagator in three
dimension is%
\begin{equation}
\sigma_{0}(x,m+B)=\frac{\exp(-(m+B)x)}{4\pi x},\sigma_{0}(p,m+B)=\frac
{1}{(m+B)^{2}+p^{2}},
\end{equation}
and its quantum correction is expressed by $\exp(A)(\mu x)^{-Cx+D}.$We find in
this expression that $\sigma(x)$ is finite provided
\begin{align}
|\sigma(x)  &  =\exp(A)\frac{\exp(-(m+B)x)}{4\pi x}(\mu x)^{-Cx+D}|=finite,\\
0  &  \leq\int_{0}^{\infty}\sigma(x)dx\leq M,\text{ }0<D,
\end{align}


and there exists $\sigma(p)$%
\begin{equation}
\sigma(p)=\int_{0}^{\infty}\frac{\sin(px)}{p}\sigma(x)dx.
\end{equation}
In this case we have not a simple pole at the Yennie gauge $d=2$ in which we
see the singularity at $p^{2}=(m+e^{2}\gamma/8\pi)^{2}.$The integral is not
anayitic and we cannot get the precise expression for $\sigma(p).$

If we expand $(\mu x)^{-Cx+D}$%
\begin{equation}
(\mu x)^{-Cx+D}=1+(-Cx+D)\ln(\mu x)+\frac{1}{2}(-Cx+D)^{2}(\ln(\mu
x))^{2}+\frac{1}{6}(-Cx+D)^{3}(\ln(\mu x))^{3}+..,
\end{equation}
term by term integral of $\sigma(p)$ converges.Using the following integrals
\begin{align}
I_{1}  &  =\int_{0}^{\infty}\frac{\sin(px)}{p}\exp(-mx)\ln(\mu x)dx\nonumber\\
&  =-\frac{\gamma}{p^{2}+m^{2}}-\frac{\ln((m^{2}+p^{2})/\mu^{2})}%
{2(p^{2}+m^{2})}-\frac{\ln((m-\sqrt{-p^{2}})/(m+\sqrt{-p^{2}}))}{p^{2}+m^{2}},
\end{align}%
\begin{align}
I_{2}  &  =\int_{0}^{\infty}\frac{\sin(px)}{p}\exp(-mx)x\ln(\mu
x)dx\nonumber\\
&  =\frac{-m}{(p^{2}+m^{2})^{2}}[\ln((m-\sqrt{-p^{2}})/(m+\sqrt{-p^{2}}%
))+\ln((p^{2}+m^{2})/\mu^{2})-2(1-\gamma)],
\end{align}
$\sigma(p)$ up to $O(e^{2})$ is given%
\begin{align}
\sigma^{(2)}(p)  &  =[\frac{1}{m^{2}+p^{2}}+\frac{A}{p^{2}+m^{2}}-\frac
{mB}{2(p^{2}+m^{2})^{2}}\nonumber\\
&  +DI_{1}-CI_{2}],p=\sqrt{-p^{2}}.
\end{align}
In this case the renormalization constant $Z$ becomes%
\begin{equation}
Z=1+A+D(-\gamma+\frac{1}{2}\ln(\frac{p^{2}+m^{2}}{\mu^{2}}))_{p^{2}%
\rightarrow\infty}\rightarrow\infty.
\end{equation}
In the Yennie gauge there remains a renormalization%
\begin{equation}
Z=1+\frac{e^{2}}{16\pi m}\ln((p^{2}+m^{2})/\mu^{2})_{p^{2}\infty}%
\rightarrow\infty.
\end{equation}
In Mikowski space, $p^{2}\rightarrow-p^{2},$with dicontinuity%
\begin{align}
\frac{1}{x-i\epsilon}  &  =P.V\frac{1}{x}+i\pi\delta(x)\\
\frac{1}{(x-i\epsilon)^{2}}  &  =P.V\frac{1}{x^{2}}+i\pi\delta^{^{\prime}%
}(x),\\
\frac{1}{x-i\epsilon}\ln(x-i\epsilon)  &  =i\pi\delta(x)\ln(x)+P.V\frac{1}%
{x}i\pi.
\end{align}
we can determine the the strucure near $p^{2}=-m^{2}$ by the imaginary part of
$\sigma(p).$Here we notice the second-order spectral function $\sigma
^{(2)}(p)$%
\begin{align}
\sigma^{(2)}(x)  &  =\frac{\exp(-mx)}{4\pi x}(1+A-Bx+(D-Cx)\ln(\mu x)),\\
\sigma^{(2)}(p)  &  =\frac{1}{m^{2}+p^{2}}+\frac{A-\gamma D}{m^{2}+p^{2}%
}-\frac{2mB}{(m^{2}+p^{2})^{2}}\nonumber\\
&  -D\frac{\ln(\sqrt{(m^{2}+p^{2})/\mu^{2}})}{(p^{2}+m^{2})}\nonumber\\
&  +\frac{Cm}{(p^{2}+m^{2})^{2}}[\ln((p^{2}+m^{2})/\mu^{2})-2(1-\gamma)].
\end{align}
Here we look at mass renormalization part
\begin{align}
Z_{m}  &  =\frac{m(2B+2C(1-\gamma)-C\ln((p^{2}+m^{2})/\mu^{2}))}{p^{2}+m^{2}%
}\nonumber\\
&  =\frac{e^{2}}{8\pi}\frac{2(d-1)-\ln((p^{2}+m^{2})/\mu^{2})}{p^{2}+m^{2}}.
\end{align}
$\operatorname{Im}\sigma^{(2)}(p)$ reads%
\begin{align}
\frac{\operatorname{Im}\sigma^{(2)}(p)}{\pi}  &  =(1+A-\gamma D)\delta
(p^{2}-m^{2})-\frac{e^{2}}{4\pi}m(d-1)\delta^{^{\prime}}(p^{2}-m^{2}%
)\nonumber\\
&  -\frac{D\theta(p^{2}-m^{2})}{m^{2}-p^{2}}+\frac{Cm\theta(p^{2}-m^{2}%
)}{(m^{2}-p^{2})^{2}}.
\end{align}
Usually we estimate the second-order spectral function in the Feynman gauge
$d=1;$%
\begin{align}
\frac{\operatorname{Im}\sigma^{(2)}(p)}{\pi}_{d=1}  &  =[(1-\frac{e^{2}}%
{8\pi\mu})\delta(p^{2}-m^{2})\nonumber\\
&  [\frac{D}{p^{2}-m^{2}}+\frac{Cm}{(m^{2}-p^{2})^{2}}]\theta(p^{2}-m^{2})].
\end{align}
This shows the renormalization constant $Z$ has an linear infrared divergence
as $\mu\rightarrow0$.If we add higher order corrections linear infrared
divergences cancell each other in the same mechanism in four
dimension[15,16,17]. In the Yennie gauge it has cuts
\begin{align}
\frac{\operatorname{Im}\sigma^{(2)}(p)}{\pi}_{d=2}  &  =\delta(p^{2}%
-m^{2})-\frac{e^{2}m}{4\pi}\delta^{^{\prime}}(p^{2}-m^{2})\nonumber\\
&  +\frac{e^{2}}{8\pi m}(\frac{1}{p^{2}-m^{2}}+\frac{2m^{2}}{(p^{2}-m^{2}%
)^{2}})\theta(p^{2}-m^{2}).
\end{align}
To see the singularity structure near $p^{2}=-m^{\ast2}$ it is better to
expand aroud $p^{2}=-m^{\ast2}.$We get%
\begin{equation}
\sigma^{(2)}(p)=[\frac{1}{m^{\ast2}+p^{2}}+(\exp(A)-1)\times(-CI_{2}%
(m\rightarrow m^{\ast})+DI_{1}(m\rightarrow m^{\ast})].
\end{equation}
In this case the renormalization constant $Z_{2}:$
\begin{equation}
Z=1+(\exp(A)-1)+D(-\gamma+\frac{1}{2}\ln(\frac{m^{\ast2}+p^{2}}{\mu^{2}}))
\end{equation}
is divergent at $p^{2}=-m^{\ast2}.$There is an interesting contribution from
$\exp(F_{2}{\normalsize )}$%
\begin{align}
&  \int_{0}^{\infty}\frac{\sin(px)}{p}\exp(-mx)(\mu x)^{D}dx\nonumber\\
&  =-\Gamma(a+1)\frac{\cos(\pi a)}{2}(p^{2}+m^{2})^{-1-a/2}\mu^{a}\nonumber\\
&  \times\frac{1}{\sqrt{-p^{2}}}[(\sqrt{-p^{2}}+m)(\frac{\sqrt{-p^{2}}%
-m}{\sqrt{-p^{2}}+m})^{-a/2}+(\sqrt{-p^{2}}-m)(\frac{\sqrt{-p^{2}}+m}%
{\sqrt{-p^{2}}-m})^{a/2}]\nonumber\\
&  \sim(\sqrt{-p^{2}}-m)^{-1-a}\text{ near }p^{2}=-m^{2},a=\frac{e^{2}}{8\pi
m}.
\end{align}


\bigskip Formally fourier transform of $\sigma(p)$ can be written by double
fourier transform%
\begin{equation}
\sigma(p)=\int\frac{d^{3}q}{(2\pi)^{3}}F.T.(\frac{\exp(-mx)}{x}(\mu
x)^{D})(q)\times F.T.(\mu x)^{-Cx}(p-q),
\end{equation}
but we do not discuss details of it in this paper.We have seen that the
fourier transformation of the propagator is very difficult.It is interesting
to study the phase structure of the model.First we estimate renormalization
constant by the following sum rule[10,11],%
\begin{equation}
Z^{-1}=\int\sigma(\omega)d\omega,
\end{equation}%
\begin{align}
Z^{-1}\int\frac{\sin(px)}{p}\frac{\exp(-m_{0}x)}{4\pi x}dx  &  =\int
\sigma(\omega)d\omega\frac{\sin(px)}{p}\frac{\exp(-\omega x)}{4\pi
x}dx,\nonumber\\
\frac{Z^{-1}}{p^{2}-m_{0}^{2}}  &  =\int\frac{\sigma(\omega)d\omega}%
{p^{2}-\omega^{2}+i\epsilon},
\end{align}%
\begin{equation}
\lim_{p\rightarrow\infty}F(p)=\lim_{p\rightarrow\infty}\int_{0}^{\infty}%
\frac{\sin(px)}{p}F(x)dx=\lim_{p\rightarrow\infty}\frac{1}{p^{2}}F(0),
\end{equation}
we can evaluate this quantity by direct substitution of $\sigma(x)$ into the
above equations and take the limit
\begin{align}
Z^{-1}  &  =\exp(A)\lim_{x\rightarrow0_{+}}\exp(-mx)(\mu x)^{-Cx+D-1}%
\nonumber\\
&  =\left[
\begin{array}
[c]{cc}%
0 & (1<D)\\
finite & (1=D)\\
\infty & (1>D)
\end{array}
\right]  .
\end{align}
We see if the coupling constant $D$ is smaller than $1,$renormalization
constant becomes $Z=0(Z^{-1}=\infty)$ and it shows a signal of confinement of
charged particle.

Next we consider the gauge dependence of the propagator[12,13,14].When we
write the photon propagator as%
\begin{equation}
D_{\mu\nu}(k)=D_{\mu\nu}^{(0)}(k)+k_{\mu}k_{\nu}M(k),
\end{equation}
we seek the change of the propagator.Under the gauge transformation defined:%
\begin{align}
\phi(x)  &  \rightarrow\exp(ie\chi(x)),\phi^{+}(x)\rightarrow\phi^{+}%
(x)\exp(-ie\chi(x)),\nonumber\\
A_{\mu}(x)  &  \rightarrow A_{\mu}(x)-\partial_{\mu}\chi,\delta(\phi
^{+}\overleftrightarrow{\partial_{\mu}}\phi)=-\partial_{\mu}\chi(\phi
^{+}\overleftrightarrow{\partial_{\mu}}\phi-2ie\phi^{+}\phi A_{\mu}),
\end{align}
propagator changes
\begin{align}
D_{\mu\nu}  &  \rightarrow i\left\langle T[A_{\mu}(x)-\partial_{\mu}^{x}%
\chi(x)][A_{\nu}(y)-\partial_{\nu}^{y}(y)]\right\rangle \nonumber\\
&  =D_{\mu\nu}+i\left\langle T\partial_{\mu}^{x}\chi(x)\partial_{\nu}%
^{y}(y)\right\rangle ,\\
\delta D_{\mu\nu}  &  =\partial_{\mu}^{x}\partial_{\nu}^{y}%
M(x-y),M(x-y)=i\left\langle T\chi(x)\chi(y)\right\rangle \nonumber\\
\Delta &  \rightarrow-i\left\langle T\phi(x)\exp(ie\chi(x)\exp(-ie\chi
(y)\phi^{+}(y)\right\rangle \nonumber\\
=  &  \Delta\left\langle T\exp(ie\chi(x))\exp(-ie\chi(y))\right\rangle .
\end{align}
Here we expand in $\delta\chi$%
\begin{align}
\left\langle T\exp(ie\delta\chi(x))\exp(-ie\delta\chi(y))\right\rangle  &
=-\frac{1}{2}e^{2}\left\langle \delta\chi^{2}(x)+\delta\chi^{2}(y)-2T\delta
\chi(x)\delta\chi(y)\right\rangle ,\nonumber\\
\delta\Delta &  =ie^{2}\Delta^{(0)}(x-y)[M(0)-M(x-y)],\nonumber\\
i[M(0)-M(x-y)]  &  =-\left\langle T\delta\chi(x)\delta\chi(y)\right\rangle .
\end{align}
Usually $M$ is identified as the gauge fixing term%
\begin{align}
M(k)  &  =-d/k^{4},\nonumber\\
ie^{2}M(x)  &  =-ie^{2}d\int\frac{d^{3}k}{(2\pi)^{3}}\frac{\exp(ik\cdot
x)-1}{k^{4}}\nonumber\\
&  =-\frac{e^{2}d}{8\pi}x.
\end{align}
We get the propagator(59) in covariant $d$ gauge for three dimension%
\begin{align}
\Delta(d,x)  &  =\exp(-\frac{e^{2}d}{8\pi}x)\Delta(0,x)\\
&  =\exp(-\frac{e^{2}d}{8\pi}x)\sigma(0,x).
\end{align}
Since $\sigma(x)$ is a full propagator it obeys the Landau-Kharatonikov
transformation in our approximation%
\begin{equation}
\sigma(d,x)=\frac{\exp(-(m^{\ast}+de^{2}/8\pi)x)}{x}\exp(A)(\mu x)^{-Cx+D}.
\end{equation}
It is not clear that the point discussed by Delbourgo,Waites[6], spectral
function in momentum space is gauge invariant for scalar which we do not
discuss here.We keep the Landau gauge and not take the covariant $d$ gauge to
avoid longitudinal photon.Here we mention the radiative correction of
mass$.$In the Landau gauge our approximation changes the mass $m$ to $m^{\ast
}=m+e^{2}(\gamma-2)/(8\pi).$The constant $\gamma$ appeared after the expansion
of $\operatorname{Ei}(1,\mu x)$ in $\mu.$In ref[5],gauge dependent self-energy
is estimated at $O(e^{2})$%
\begin{equation}
\Sigma(m)=d\frac{e^{2}}{4\pi},
\end{equation}
and the physical mass for fermion defined in the Landau gauge
\begin{equation}
m_{phy}=m+\Sigma(m).
\end{equation}
If we estimate one-loop self-energy in the Landau gauge
\begin{equation}
\Sigma(m)=\frac{e^{2}}{2\pi},
\end{equation}
which is two-times larger than $e^{2}/(4\pi)$ in $m^{\ast}$ since we used
$D_{+}(x)$ instead of $D(x).$We see the same effects occured for scalar with
vertex correction.We call $m^{\ast}=m+e^{2}(\gamma-2)/8\pi$ the renormalized
mass in our approximation.At first sight, mass shift seems to be peculiar but
infrared behaviour is governed by one-particle intermediate state.

\section{Spinor QED}

In this section we study the spectral function for massive fermion.Similar to
scalar case propagator and the spectral function are defined%
\begin{align}
S(p)  &  =\int d^{3}x\exp(ip\cdot x)\left\langle \Omega|T(\psi(x)\overline
{\psi}(0))|\Omega\right\rangle \nonumber\\
&  =(\int_{m}^{\infty}+\int_{-\infty}^{-m})\frac{\rho(\omega)d\omega}%
{p\cdot\gamma-\omega+i\epsilon},\\
\rho(p^{2})  &  =(2\pi)^{2}\sum_{N}\delta^{(3)}(p-p_{N})\left\langle
\Omega|\psi|N\right\rangle \left\langle N|\overline{\psi}|\Omega\right\rangle
.
\end{align}


Matrix element is%
\begin{align}
T_{n}  &  =\left\langle \Omega|\psi|r;k_{1},..,k_{n}\right\rangle ,\\
T_{n}^{\mu}  &  =-\int d^{3}x\exp(ik_{n}\cdot x)\left\langle \Omega|T\psi
j^{\mu}|r;k_{1},...k_{n-1}\right\rangle ,
\end{align}
provided%
\begin{equation}
\square_{x}T\psi A_{\mu}(x)=T\psi\square_{x}A_{\mu}(x)=T\psi(-j_{\mu
}(x)+\partial_{\mu}^{x}(\partial\cdot A(x))).
\end{equation}
In the similar way to the scalar case $T_{n}$ satisfies Ward-Identity:
\begin{align}
\partial_{\mu}^{x}T(\psi j_{\mu}(x))  &  =-e\psi(x),\\
\partial_{\mu}^{x}T(\overline{\psi}j_{\mu}(x))  &  =e\overline{\psi}(x),\\
k_{n\mu}T_{n}^{\mu}(r,k_{1},k_{2},..k_{n})  &  =eT_{n-1}(r,k_{1}%
,k_{2},..k_{n-1}),r^{2}=m^{2}.
\end{align}
One photon matrix element by LSZ is
\begin{align}
T_{1}  &  =\left\langle in|T(\psi_{in}(x),ie\int d^{3}y\overline{\psi}%
_{in}(y)\gamma_{\mu}\psi_{in}(y)A_{in}^{\mu}(y))|r;k\text{ }in\right\rangle
\nonumber\\
&  =ie\int d^{3}yd^{3}zS_{F}(x-y)\gamma_{\mu}\delta^{(3)}(y-z)\exp(i(k\cdot
y+r\cdot z))\epsilon^{\mu}(k,\lambda)U(r,s)\nonumber\\
&  =-ie\frac{(r+k)\cdot\gamma+m}{((r+k)^{2}-m^{2})}\gamma_{\mu}\epsilon^{\mu
}(k,\lambda)\exp(i((k+r)\cdot x))U(r,s),
\end{align}
where $U(r,s)$ is a two-component free particle spinor with positive energy:%
\begin{equation}
\sum_{S}U(r,s)\overline{U}(r,s)=\frac{\gamma\cdot r+m}{2m}.
\end{equation}
In this case the function $F$ becomes%
\begin{align}
F  &  =e^{2}\int\frac{d^{3}k}{(2\pi)^{3}}\exp(ik\cdot x)\theta(k^{0}%
)\delta(k^{2})tr[\frac{(r+k)\cdot\gamma+m}{((r+k)^{2}-m^{2})}\gamma_{\mu}%
\frac{\gamma\cdot r+m}{2m}\gamma_{\nu}\frac{(r+k)\cdot\gamma+m}{((r+k)^{2}%
-m^{2})}\Pi^{\mu\nu}]\nonumber\\
&  =-e^{2}\int\frac{d^{3}k}{(2\pi)^{3}}\exp(ik\cdot x)\theta(k^{0}%
)[\delta(k^{2}-\mu^{2})(\frac{m^{2}}{(r\cdot k)^{2}}+\frac{1}{(r\cdot
k)})+(d-1)\frac{\partial\delta(k^{2}-\mu^{2})}{\partial k^{2}}].
\end{align}%
\begin{equation}
F=\frac{e^{2}m^{2}}{8\pi r^{2}}(-\frac{1}{\mu}+x(1-\gamma)-x\ln(\mu
x))+\frac{e^{2}}{8\pi r}(\ln(\mu x)+\gamma)+\frac{e^{2}}{8\pi}(d-1)(\frac
{1}{\mu}-x).
\end{equation}
Thus the lowest order spectral function is exactly the same with that in the
scalar case except for the normalization factor in the phase space integral:%

\begin{align}
\rho(p^{2}) &  =\int\frac{d^{3}x}{(2\pi)^{3}}\exp(ip\cdot x)\rho(x)\nonumber\\
&  =\int\frac{d^{3}x}{(2\pi)^{3}}\exp(ip\cdot x)\int d^{2}r\frac{m}{r_{0}}%
\exp(ir\cdot x)\exp(F).
\end{align}
After angular integral we get
\begin{equation}
\rho(p)=\frac{m}{4\pi p}\exp(A)\int_{0}^{\infty}dx\sin(px)\frac{\exp
(-(B+m)x)}{x}(\mu x)^{-Cx+D}.
\end{equation}
Therefore the structure is the same with scalar case as we mentioned in the
last section. Consequently there is no infrared divergences at $p^{2}=m^{2}.$

For the renormalization constants we apply the same argument as for the
scalar,full propagator is expressed in the dipersion integral%
\begin{align}
S_{F}(x) &  =-\int\rho(\omega)d\omega(i\gamma\cdot\partial+\omega)\frac
{\exp(-\omega x)}{4\pi x}=-(i\gamma\cdot\partial\rho(x)+\rho(x)),\\
\frac{Z_{2}^{-1}}{\gamma\cdot p-m_{0}} &  =\int\frac{\rho(\omega)d\omega
}{\gamma\cdot p-\omega+i\epsilon}\\
Z_{2}^{-1} &  =\int\rho(\omega)d\omega=\exp(A)\lim_{x\rightarrow0_{+}}(\mu
x)^{-Cx+D-1},\\
m_{0}Z_{2}^{-1} &  =\int\omega\rho(\omega)d\omega=\exp(A)\lim_{x\rightarrow
0_{+}}(\mu x)^{-Cx+D-1}\nonumber\\
&  =\exp(A)\mu^{D}\lim_{x\rightarrow0_{+}}(x^{-Cx+D-1})=\left[
\begin{array}
[c]{cc}%
0 & (1<D)\\
finite & (1=D)\\
\infty & (1>D)
\end{array}
\right]  .
\end{align}
Therefore there is a confining phase;$Z_{2}=0(Z_{2}^{-1}=\infty)$ of charged
particle for weak coupling constant$.$This fact also implies the vanishment of
the bare mass in renormalization theory%
\begin{equation}
\frac{m_{0}}{m}=Z_{2}=0.
\end{equation}
Order parameter for the vacuum expectation value of pair condensate is given
\begin{align}
\left\langle \overline{\psi}\psi\right\rangle  &  =-itrS_{F}(x)=-2m\mu^{D}%
\lim_{x\rightarrow0_{+}}(\exp(-mx)(x)^{-Cx+D-1}))\nonumber\\
&  =\left[
\begin{array}
[c]{cc}%
0 & (1<D)\\
finite & (1=D)\\
\infty & (1>D)
\end{array}
\right]  .
\end{align}
Here we notice that there is a critical coupling constant $D_{cr}%
=1(e^{2}/(8\pi m)=1).$

\section{Summary}

We have seen how the Ward-Identity for soft photon may be applied to
three-dimensional electro-dynamics to extract full propagator in position
space and the infrared behaviour of the propagator in momentum
space.Neglecting unconventional terms in the elecrtomagnetic current for
scalar,spectral functions for scalar and spinor coincide each
other.Exponentiation of the lowest order spectral function corresponds to the
infinite ladder approximation. Since the theory is super renormalizable the
mass are corrected to add some finite radiative correction.Logarithmic mass
renormalization alters the structure of the propagator and is essencially
non-pertubative.Wave functuin renormalization modifies the canonical dimension
of the propagator.This effect is familiar in dimensional annalysis.These are
the consequences of logarithmic infrared divergences. Renormalization constant
vanishes for both scalar and spinor cases and there is a confining phase for
weak coupling.This picture is consistent with perturbative logarithmic
infrared divergences at $p^{2}=m^{2}.$If the coupling constant is smaller than
$D_{cr}$ ($D=e^{2}/8\pi m\leq1),$vacuum expectation value $\left\langle
\overline{\psi}\psi\right\rangle $ becomes finite.In our lowest order spectral
function there remains linear infrared divergences that was regularized by
photon mass.$O(e^{2})$ correction to the bare vertex and the external line and
the problems of cancellation of infrared divergences are now in progress[17].

\section{Aknowledgement}

The author would like thank Prof.Roman Jackiw to his introduction on technique
based on Ward-Identity at MIT September 2002.

\section{Appendix}

{\normalsize To evaluate }$F$ we use {\normalsize the function }$D_{+}(x)$ in
three dimension:%

\begin{align}
D_{+}(x)  &  =\frac{1}{(2\pi)^{2}i}\int\exp(ik\cdot x)\theta(k^{0}%
)\delta(k^{2}-\mu^{2})d^{3}k\nonumber\\
&  =\frac{1}{(2\pi)^{2}i}\int_{0}^{\infty}J_{0}(kx)\frac{\pi kdk}{2\sqrt
{k^{2}+\mu^{2}}}=\frac{\exp(-\mu x)}{8\pi ix},
\end{align}
{\huge \qquad}

with the following parameter trick as in ref[9]%

\begin{align}
\lim_{\epsilon\rightarrow0_{+}}\int_{0}^{\infty}d\alpha\exp(i(k+i\epsilon
)\cdot(x+\alpha r))  &  =\frac{\exp(ik\cdot x)}{k\cdot r},\\
\lim_{\epsilon\rightarrow0_{+}}\int_{0}^{\infty}\alpha d\alpha\exp
(i(k+i\epsilon)\cdot(x+\alpha r))  &  =\frac{\exp(ik\cdot x)}{(k\cdot r)^{2}}.
\end{align}
The function $F$ is written in terms of the parameter integrals
\begin{align}
F  &  =ie^{2}m^{2}\int_{0}^{\infty}\alpha d\alpha D_{+}(x+\alpha r,\mu
)-e^{2}\int_{0}^{\infty}d\alpha D_{+}(x+\alpha r,\mu)\nonumber\\
&  -ie^{2}(d-1)\frac{\partial}{\partial\mu^{2}}D_{+}(x,\mu)\nonumber\\
&  =\frac{e^{2}m^{2}}{8\pi r^{2}}(-\frac{\exp(-\mu x)}{\mu}+x\operatorname{Ei}%
(1,\mu x))-\frac{e^{2}}{8\pi r}\operatorname{Ei}(1,\mu x)+(d-1)\frac{e^{2}%
}{8\pi\mu}\exp(-\mu x),
\end{align}
where the function $\operatorname{Ei}(n,x)$ is defined
\begin{equation}
\operatorname{Ei}(n,x)=\int_{1}^{\infty}\frac{\exp(-xt)}{t^{n}}dt.
\end{equation}
It is understood that all terms which vanish with $\mu\rightarrow0$ are
ignored.
\begin{equation}
\operatorname{Ei}(1,\mu x)=-\gamma-\ln(\mu x)+O(\mu x),
\end{equation}%
\begin{align}
F_{1}  &  =\frac{e^{2}m^{2}}{8\pi r^{2}}(-\frac{1}{\mu}+x(1-\ln(\mu
x)-\gamma))+O(\mu),\\
F_{2}  &  =\frac{e^{2}}{8\pi r}(\ln(\mu x)+\gamma)+O(\mu),\\
F_{g}  &  =\frac{e^{2}}{8\pi}(\frac{1}{\mu}-x)(d-1)+O(\mu).
\end{align}
Here $\gamma$ is Euler's constant.Using the integrals%
\begin{equation}
\int d^{3}x\exp(ip\cdot x)\int d^{3}r\delta(r^{2}-m^{2})\exp(ir\cdot
x)f(r)=f(m),
\end{equation}%
\begin{equation}
\int d^{3}x\exp(ip\cdot x)\int\frac{d^{3}r}{(2\pi)^{3}}\delta(r^{2}%
-m^{2})=\frac{1}{m^{2}+p^{2}}.
\end{equation}
we get $F$ in position space%
\begin{equation}
F=\frac{e^{2}}{8\pi\mu}(d-2)+\frac{\gamma e^{2}}{8\pi r}+\frac{e^{2}}{8\pi
r}\ln(\mu x)-\frac{e^{2}}{8\pi}x\ln(\mu x)-\frac{e^{2}}{8\pi}x(d+\gamma-2).
\end{equation}
After integration over $r$
\begin{align}
\int_{0}^{\pi}\exp(ir\cdot x\cos(\theta))d\theta &  =\pi J_{0}(rx),\\
\int_{0}^{\infty}dr\frac{\pi rJ_{0}(rx)}{\sqrt{r^{2}+m^{2}}}  &  =\frac
{\exp(-mx)}{x},
\end{align}
we obtain the full propagator%
\begin{align}
\sigma(x)  &  =\exp(A)\frac{\exp(-(m+B)x)}{4\pi x}(\mu x)^{-Cx+D},\\
A  &  =\frac{e^{2}}{8\pi\mu}(d-2)+\frac{\gamma e^{2}}{8\pi m},B=\frac{e^{2}%
}{8\pi}(d+\gamma-2),C=\frac{e^{2}}{8\pi},D=\frac{e^{2}}{8\pi m}.
\end{align}


\section{References}

[1] R.Jackiw,S.Templeton,Phys.Rev.\textbf{23D}.2291(1981)

[2] E.I.Guendelman,Z.M.Raulvic,Phys.Rev.\textbf{30D}.1338(1984)

[3] T.Appelquist,D,Nash,L.C.R.Wijewardhana,Phys.Rev.Lett.\textbf{60}.2575(1988)

[4] Y.Hoshino,T.Matsuyama,Phys.Lett.\textbf{222B}.493(1989)

[5] S.Deser,R.Jackiw,S.Templeton,Ann.Phys.\textbf{140}.372(1982)

[6] A.B.Waites,R.Delbourgo,Int.J.Mod.Phys.\textbf{27A}.6857(1992)

[7] D.Atkinson,D.W.E.Blatt,Nucl.Phys.\textbf{151B}.342(1979)

[8] P.Maris,Phys.Rev.\textbf{52D}.6087(1995);Y.Hoshino,IL Nuovo
Cim.\textbf{112A}.335(1999)

[9] R.Jackiw,L.Soloviev,Phys.Rev\textbf{173}.1458(1968)

[10] R.Delbourgo,IL Nuovo Cim.\textbf{49A}.485(1978)

[11]K.Nishijima,Prog.Theor.Phys;\textbf{81}.878(1989),ibid.\textbf{83}.1200(1990)

[12]L.D.Landau,J,M.Kharatonikov,Zk.Eksp.Theor.Fiz.\textbf{29}.89(1958)

[13]B.Zumino,J.Math,Phys.\textbf{1}.1(1960)

[14] in Landau,Lifzhits,Quantum Electrodynamics (Butterworth Heinemann)

[15]T.Kinoshita,Prog.Theor.Phys.\textbf{5},1045(1950)

[16]N.Nakanishi,Prog.Theor.Phys.\textbf{19},159(1958)

[17] Y.Hoshino,in preparation


\end{document}