\documentstyle[epsf,amsmath]{article}
\renewcommand{\thefootnote}{\*{*}}
\newcommand{\1}{{\sf 1 \!\! 1}}
\newcommand{\lh}[1]{\slash \! \! \!  #1}
\newcommand{\simgt} 
{\mbox{\raisebox{-0.5ex}{$ \; \sim$}
\raisebox{0.8ex}{$ \!\!\!\!\!\!\! >$}}}
\textwidth 162mm \textheight 235mm \topmargin -10mm
\oddsidemargin 0mm
\addtolength{\textheight}{-\headheight}
\addtolength{\textheight}{-\headsep}
\addtolength{\textheight}{-\footheight}
\parindent 0mm
\setlength{\parskip}{\baselineskip}
\thispagestyle{empty}
\pagenumbering{arabic}
\begin{document}
\setcounter{page}{0}
\mbox{ }
\rightline{UCT-TP-260/03}\\
\rightline{February 2003}\\
\vspace{3.5cm}
\begin{center}
{\LARGE \bf Electromagnetic nucleon form factors from\\
 QCD sum rules}\footnote{Work supported in part by Fondecyt
grants No.1010976 \\
\vspace{.5cm}
{\Large H. Castillo $^{(a),(b)}$ , C.A. Dominguez $^{(c)}$, 
M. Loewe $^{(a)}$}\\
\vspace{.4cm}
\end{center}
%\begin{center}
(a) Facultad de F\'{\i}sica, Pontificia Universidad Cat\'{o}lica  de Chile, 
Casilla 306, Santiago 22, Chile.\\
(b) Departamento  de 
Ciencias,  Pontificia Universidad Cat\'{o}lica del Per\'{u}, Apartado 1761,
 Lima, Per\'{u}.\\
(c) Department of Physics,  University of Cape 
Town, Rondebosch 7701, South Africa.\\
%\end{center}
\vspace{.5cm}
\begin{abstract}
\noindent
The electromagnetic form factors of the nucleon,
in the space-like region, are determined from 
three-point function Finite Energy QCD Sum Rules.
The QCD calculation is performed to leading order
in perturbation theory (in the chiral limit), and to 
leading order in the non-perturbative power corrections,
proportional to the quark condensate. The results are
in  good agreement with the data for both the proton
and neutron form factors in
the currently accessible experimental region 
of momentum transfers.
\end{abstract}
\newpage
\setlength{\baselineskip}{1.5\baselineskip}
\noindent

The asymptotic behaviour of the electromagnetic form factors of the nucleon 
have been studied in perturbative QCD, together with QCD sum rule estimates 
of the nucleon wave functions \cite{REV1}.  Comparison with data is 
difficult due to the extreme asymptotic
nature of these results.  In addition, these wave functions are affected by 
some unavoidable model dependency. In view of this,  it is desirable to 
attempt a  field theoretic QCD determination in a region of experimentally 
accessible momentum transfers , and without  any reference to the concept of 
a wave function. \\
In this note we determine the electromagnetic nucleon form factors, in a 
wide range of (space-like) momentum transfers, in the framework of 
three-point function Finite Energy QCD sum rules. As is well known by now, this
technique  is based on the Operator Product Expansion (OPE) of current 
correlators at  short distances, and on the notion of quark-hadron duality
\cite{QCDSR}. 
Analyticity and dispersion relations connect then the QCD information in the 
OPE with hadronic parameters entering the corresponding spectral functions. 
We compute the QCD correlator  to leading order in perturbative QCD, in the 
chiral limit ($m_u=m_d=0$), and include the leading order non-perturbative 
power correction proportional to the quark-condensate.\\
We begin by considering the following three-point function (see Fig. 1)\\
%Eq.1
\begin{equation}
\Pi_\mu (p^2,p'^{2},Q^2) =  i^2 \!\int \!\!\mbox{d}^ 4 x \!\!\int
 \!\!\mbox{d}^4 y \; e
^{i (p' \cdot x - q  \cdot y)}\, \langle 0 \left| T \{ \eta_N (x)
 J_\mu^{EM}(y)\bar
 \eta_N  
(0) \}
\right| 0\rangle \; , 
\end{equation}

where $Q^2\equiv - q^2 = -(p' - p)^2\geq 0$ is fixed, and
%Eq.2
\begin{equation}
\eta_N(x)=\varepsilon_{abc}\left[u^a(x)(C\gamma_\alpha)u^b(x)\right]
(\gamma^5\gamma^\alpha d^c(x)) \label{ec2}
\end{equation}
is an interpolating current with nucleon (proton) quantum numbers; the 
neutron case $u \leftrightarrow  d$ will be discussed at the end.
In Eq.(1),  $J^{\mu}_{EM}$ is the electromagnetic current\\
%Eq3
\begin{equation}
J_{EM}^\mu (y)=\frac 2 3 \bar u(y)\gamma^\mu u(y)-\frac 1 3 \bar 
d(y)\gamma^\mu d(y) \; .
\end{equation}
The current Eq.(2) couples to a nucleon of momentum $p$ and polarization $s$ 
according to
%Eq.4
\begin{equation}
\langle 0\left| \eta_{N}(0) \right|N(p,s)\rangle = \lambda_N u(p,s) ,
\end{equation}
where $u(p,s)$ is the nucleon spinor, and $\lambda_N$, the current-nucleon 
coupling, is a phenomenological parameter a-priori unknown. This parameter 
can be estimated, e.g. using QCD sum rules for a two-point function 
involving the currents $\eta_N$ \cite{MN}. In this case one can determine 
the nucleon mass, as well as the coupling $\lambda_N$.\\

Concentrating first on the hadronic sector, and inserting a one-particle 
nucleon state in the three-point function (1) brings out the nucleon form 
factors $F_1(q^2)$, and $F_2(q^2)$,
defined as
%Eq.5
\begin{equation}
\langle k_1\, s_1\left|  J_\mu^{EM}(0) \right| k_2\, s_2\rangle=
\bar u_N (k_1,s_1)
\left[ F_1 (q^2) \gamma _\mu + \frac{i \kappa}{2M_N}
F_2(q^2)\sigma_{\mu \nu} q^\nu \right]  
u_N(k_2,s_2) \; ,
\end{equation}
where $q^2= (k_2-k_1)^2$, and $\kappa$ is the anomalous magnetic moment
in units of nuclear magnetons ($\kappa_p = 1.79$ for  the proton, and
$\kappa_n = - 1.91$  for the neutron). The form factors 
$F_{1,2}(q^2)$ are related to the electric and magnetic (Sachs) form factors 
$G_E(q^2)$, and $G_M(q^2)$, measured in elastic electron-proton scattering 
experiments, according to
%Eq.6 and Eq.7
\begin{eqnarray}
G_E(q^2) &\equiv& F_1(q^2)+\frac{\kappa q^2}{(2m)^2} F_2(q^2)\; ,
\\ [.2cm]
G_M(q^2) &\equiv& F_1(q^2)+ \kappa F_2(q^2) \; ,
\end{eqnarray}
where $G_E^p(0) = 1$, $G_M^p(0) = 1 + \kappa_p$ for the proton, and
$G_E^n(0) = 0$,  $G_M^n(0) = \kappa_n$ for the neutron.
Next, the hadronic spectral function is obtained after inserting a complete 
set of  nucleonic states in (1), and computing the discontinuity in the 
complex $p^2 \equiv s$, ${p'}^2 \equiv s'$
plane. For $s, s' < 2.1\, \mbox{GeV}^2$, i.e. below the Roper resonance, one 
can safely approximate the hadronic spectral function by the single-particle 
nucleon pole, followed by a continuum with thresholds $s_0$ and $s'_0$ 
($s_0, s'_0 > M_N^2$). This hadronic continuum is expected to coincide 
numerically with the perturbative QCD (PQCD) spectral function (local
duality). 
This procedure is standard in QCD sum rule applications, and leads to
%Eq.8
\begin{equation}
\begin{aligned}
\mbox{Im} \Pi_\mu(s,s',Q^2)\Big|_{HAD}&=\pi^2\,\lambda_N^2\, \delta
(s-M_N^2)
\delta(s' - M_N^2) \\
 & \times \Bigg \{ F_1(q^2) \left[\lh p' \gamma_\mu \lh p+M_N (\lh p' \gamma_\mu+
\gamma_\mu \lh p)+ M_N^2 \gamma_\mu \right]\\ &+\frac{i \kappa}{2
M_N}F_2(q^2)\left[\lh p' \sigma_{\mu\nu}\lh p+M_N(\lh p'
\sigma_{\mu\nu}+\sigma_{\mu\nu}\lh p)+M_N^2 \sigma_{\mu\nu} 
\right] q^\nu
\Bigg \} \Theta\left(s_0-s\right)  \\
&+\mbox{Im} \Pi_{\mu}(s,s',Q^2)\Big|_{PQCD}\Theta\left( s-s_0\right)
\; ,
\end{aligned} 
\end{equation}
where we have set $s_0=s'_0$ for simplicity.\\

Turning to the QCD sector, the three-point function (1) to leading order in 
perturbative
QCD, and in the chiral limit, is given by\\
%Eq.9
\begin{equation}
\begin{aligned}
\Pi^\mu(p^2,{p'}^2,Q^2)= 16 \int \mbox{d}^ 4 x \; \int \mbox{d}^ 4 y 
\; e^{i(p' \cdot x-q
\cdot y)}\; \mbox{Tr}\left[\int\frac {\mbox{d}^4 {k_1}} {(2\pi)^4} 
\frac {\lh {k_1}}{k_1^2}
e^{-i k_1 \cdot (x-y)}\gamma^\mu \; \times \right. \\ \left. \int\frac 
{\mbox{d}^4 
{k_2}} {(2\pi)^4} \frac {\lh {k_2}}{k_2^2} e^{-i k_2 \cdot y} \gamma_\nu 
\int\frac {\mbox{d}^4 {k_3}} {(2\pi)^4} \frac {\lh {k_3}}{k_3^2}
e^{+i k_3 \cdot x} \gamma_\alpha\right]\left(\gamma^5\gamma^\alpha 
\int\frac {\mbox{d}^4 {k_4}} {(2\pi)^4} \frac {\lh {k_4}}{k_4^2}
e^{-i k_4 \cdot x}
\gamma^\nu\gamma^5\right)\\
-4 \int\mbox{d}^4 x\int \mbox{d}^4 y \;e^{i( p' \cdot x-q \cdot y)} \;
\mbox{Tr}\left[
\int\frac {\mbox{d}^4 {k_4}} {(2\pi)^4} \frac {\lh {k_4}}{k_4^2}
e^{-i k_4 \cdot x} \gamma_\nu 
\int\frac {\mbox{d}^4 {k_3}} {(2\pi)^4} \frac {\lh {k_3}}{k_3^2}
e^{+i k_3 \cdot x} \gamma_\alpha\right]\\
\times \left(\gamma^\alpha 
\int\frac {\mbox{d}^4 {k_1}} {(2\pi)^4} \frac {\lh {k_1}}{k_1^2}
e^{- i k_1 \cdot (x-y)}  \gamma^\mu 
\int\frac {\mbox{d}^4 {k_2}} {(2\pi)^4} \frac {\lh {k_2}}{k_2^2}
e^{-i k_2 \cdot y} \gamma^\nu\right).
\end{aligned}
\end{equation}

After computing the traces and performing the momentum space integrations, 
Eq.(9) involves several Lorentz structures analogous to those entering
the hadronic spectral function  Eq. (8).
Before invoking duality one needs to choose a particular Lorentz structure 
present in both (8) and (9). A convenient choice turns out to be 
$\lh{p'} \gamma_\mu  \lh{p}$, which
allows to project $F_1(q^2)$, as this structure does not appear multiplying 
$F_2(q^2)$ in Eq.(8). An additional advantage of this choice is that the 
quark condensate  contribution, to be discussed later, does not involve the
structure  $\lh{p'} \gamma_\mu \lh{p}$, on account
of vanishing traces. Hence, $F_1(q^2)$ will be dual only to the PQCD  
expression. It must be pointed  out that the PQCD spectral function contains
the structure 
$\lh{p'} \gamma_\mu \lh{p}$ explicitly, as well as implicitly,  i.e. 
there are terms proportional to this structure which are generated only 
after momentum space integration.\\
After a very lengthy calculation, the imaginary part of Eq.(9) is given by
%Eq.10
\begin{equation}
\mbox{Im} \Pi^\mu(s,s',Q^2) =\left[\frac 4 {(2\pi)^8}(3
\Omega_1+ 4 \Omega_2-\Omega_3)\right]\left(\lh p' \gamma^\mu
\lh p\right)+\ldots \; ,
\end{equation}
where
%Eq.11-Eq.13
%Eq.11
\begin{eqnarray}
\Omega_1&=&\frac{{\pi }^6} 2 \,\left( Q^2 + s - s' -  \frac{Q^4 + 
2\,Q^2\,s +
s^2 - 2\,s\,s' -  {s'}^2}{{\sqrt{Q^4 +
          {\left( s - s' \right) }^2 +
          2\,Q^2\,\left( s + s' \right) }}} \right)\; , \\ 
          [.7cm]
%Eq.12
\Omega_2&=&{\pi}^6 \left\{\frac{\left( 2\,Q^2 + 3\,s - 3\,s' \right) }{3}
-\frac{\left[( Q^2 + s)^3( 2\,Q^2 + 3s) + 
       3\,\left( Q^6 - 5Q^2s^2 - 4s^3 \right)s'\right]}
 {3{\left[ Q^4 + {( s - s')}^2 + 
         2\,Q^2( s + s')  \right] }^{\frac{3}{2}}} \right.\nonumber  \\[.2cm]
 &+& \left. \frac{ \left[ (3Q^2 - 4s) ( Q^2 + 3s){s'}^2 + 7Q^2{s'}^3 + 
       3{s'}^4 \right]} {3\left[ Q^4 + {( s - s')}^2 + 
         2\,Q^2( s + s')  \right] ^{\frac{3}{2}}}\right\} \; ,  \\
         [.9cm]
%Eq.13
\Omega_3 &=&  \frac{-\left\{ \pi ^6\,\left[ 23\,Q^2 + 18\,\left( -s + s'
\right) \right]
\right\} }{72} 
+\frac{\pi^6}{72\,\left[ Q^4 + \left( s - s' \right) ^2 + 2\,Q^2\,
\left( s + s' \right)  \right] ^
\frac{5}{2}} \nonumber \\[.2cm]
&\times&  \left[ \left(23\,Q^2 - 18\,s \right) \,\left( Q^2 +
 s \right)^5  
+ \left( Q^2 + s \right) ^3\,\left( 133\,Q^4 - 169\,Q^2\,s + 108\,s^2
 \right) \,s' \right. \nonumber \\ 
&+& \left. 2\,\left( 160\,Q^8 + 6\,Q^6\,s + 3\,Q^4\,s^2 + 40\,Q^2\,s^3 - 117\,
s^4 \right)
\,{s'}^2  \right. \nonumber\\   
&+& \left. 2\,\left( 205\,Q^6 - 61\,Q^4\,s - 122\,Q^2\,s^2 + 108\,s^3 \right)
 \,{s'}^3  \right. \nonumber \\ 
&+& \left.  \left( 295\,Q^4 - 37\,Q^2\,s - 54\,s^2 \right) \,{s'}^4 +
 \left( 113\,Q^2 - 36\,s \right) \,{s'}^5 + 18\,{s'}^6  \right] \;.
\end{eqnarray}

Equation (11) corresponds to the terms containing  $\lh{p'} \gamma_\mu 
\lh{p}$ explicitly, and Eqs.(12)-(13) to the implicit case. The spectral 
function (10)  contains
additional terms proportional to other (independent) Lorentz structures, 
which are not given. Collecting all three terms in (10) leads to
%Eq.14
\begin{eqnarray}
\mbox{Im} \Pi^\mu(s,s',Q^2)&=&
\frac{323\,Q^2 + 378\,\left( s - s' \right) }{4608\,{\pi }^2} \;
+ \frac 1 {4608\,\pi ^2\,
\left[ Q^4 + ( s - s') ^2 + 2\,Q^2( s + s')\right]^{\frac{5}{2}}}
\nonumber \\ [.2cm]
&\times&
\left[ -323\,Q^{12} - Q^{10}( 1993\,s + 1237\,s')  
- 10\,Q^8( 512\,s^2 + 323\,s\,s' + 134\,{s'}^2)\right. \nonumber \\
&+&\left.  Q^6 ( -7010\,s^3 + 1188\,s\,{s'}^2 + 550\,{s'}^3 )
\right. \nonumber\\
&+& \left.  Q^4( -5395s^4 + 7010s^3s' + 2610s^2{s'}^2 + 3146s{s'}^3 
+2165{s'}^4) \right. \nonumber \\
&-& \left. Q^2\,( s - s') ^2 (2213\,s^3 - 2859\,s^2\,s' - 
3099\,s\,{s'}^2 - 1567\,{s'}^3) \right. \nonumber \\
&-&\left. 378\,( s - s')^4 (s^2 - 2\,s\,s' - {s'}^2 )\right]
\;\lh p'\gamma^\mu \lh p+\ldots
\end{eqnarray}

The next step is to invoke  (global) quark-hadron duality,  according to 
which the area under the hadronic spectral function equals the  area under 
the  corresponding QCD
spectral function. The integrals in the complex energy plane may involve any 
analytic integration kernel; this leads to different kinds of QCD sum rules, 
e.g. Laplace (negative exponential kernel), Finite Energy Sum Rules (FESR) 
(power kernel), etc. We choose the latter, as they have the advantage of 
being organized according to dimensionality (to leading order in gluonic 
corrections to the vacuum condensates). In this case the leading dimension
FESR is
%Eq.15
\begin{equation}
\int_0^{s_0}\mbox{d} s
\int_0^{s_0-s} \mbox{d} {s'}\;\mbox{Im}\Pi(s,s',Q^2)\mid_{HAD}=
 \int_0^{s_0}
\mbox{d} s
\int_0^{s_0-s} \mbox{d} {s'}\; \mbox{Im}\Pi(s,s',Q^2)\mid_{QCD} 
\; .
\end{equation}
The integration region, shown in Fig. 2, has been chosen as a triangle; the 
main contribution being that of region I, and the area included from regions
II and III tends  to compensate
the excluded regions. Other choices, e.g. rectangular regions, lead to 
similar final results, as discussed in \cite{IOFFE81}-\cite{DLR}. After 
performing the integrations, one finally obtains
%Eq.16
\begin{eqnarray}
F_1(Q^2)&=&\frac{2\,{s_0}\,\left( 96\,Q^6 + 297\,Q^4\,{s_0} + 158\,Q^2\,
{{s_0}}^2 -
112\,{{s_0}}^3 \right)} {9216\,{\pi }^4\,\left( Q^2 + 2\,{s_0} \right) \,
    {{{\lambda }_N}}^2} \nonumber \\ [.2cm]
 &+& \frac{ 
    3\,\ln (\frac{Q^2}{Q^2 + 2\,{s_0}})\,\left(Q^2 + 2\,{s_0} \right) \,
\left( 32\,Q^6 + 67\,Q^4\,{s_0} + 7\,Q^2\,{{s_0}}^2 - 21\,{{s_0}}^3
 \right) }
{9216\,{\pi }^4\,\left( Q^2 + 2\,{s_0} \right) \,
    {{{\lambda }_N}}^2}  \; ,
\end{eqnarray}

where one can recognize the standard logarithmic singularity arising from 
the chiral limit.\\

We now turn to the extraction of $F_2(q^2)$, and consider the leading order 
non-perturbative power correction to the OPE, in this case given by the 
quark condensate contribution. It turns out that the contribution involving
the up-quark condensate vanishes (on account of vanishing traces),
leaving only the piece proportional to 
$\langle \bar{d} d \rangle$. The three-point function (1) becomes (see 
Fig. 3)
%Eq.17
\begin{eqnarray}
&&{\Pi^{\langle \bar q q\rangle}}^\mu(p^2,{p'}^2,Q^2)= i \frac { 
{\langle \bar d d\rangle} }
{3 (2 \pi)^4} \left[  4 \,\int \!
\mbox{d}^4 k \frac {\mbox{Tr} \left[\lh k \gamma^\mu (\lh k-\lh q )
\gamma_\nu (\lh k- \lh p')\gamma_\alpha \right] } {(k-q)^2 (k-p')^2 k^2}\,
 \gamma^
\alpha\gamma^\nu \right. \nonumber \\ [.2cm]
&-& \left. \int \!\mbox{d}^4 {k}
\frac{\mbox{Tr}\left[\lh {k}\gamma_\nu (\lh {k}-\lh { p'})
\gamma_\alpha\right]}{k^2 (k-p')^2 q^2} \left(\gamma^\alpha\gamma^\mu
\lh q \gamma^\nu \right)
+\int\!\mbox{d}^4
 k
\frac{\mbox{Tr}\left[\lh k \gamma_\nu (\lh k-\lh p)\gamma_\alpha\right]}
{k^2 (k-p)^2 q^2} (\gamma^\alpha \lh q \gamma^\mu \gamma ^\nu)\right] \; .
\end{eqnarray}

Our choice of Lorentz structure in this case is $\lh{q} \gamma^\mu$, 
which appears in Eq.(17), as well as in Eq.(8) where it multiplies  
$F_2(q^2)$, but not $F_1(q^2)$. In fact, after some algebra
%Eq.18
\begin{equation}
\mbox{Im} {\Pi^{\langle \bar d d\rangle}}^\mu(s,s',Q^2)
\Big|_{\mbox{QCD}}
= -\frac {\langle \bar d d\rangle} {3}\left\{
 \frac{ Q^2 s' \left( Q^2 + 3 s + s'\right) }
  {\left[ Q^4 + \left( s - s' \right)^2 + 2Q^2\left( s + s' \right)
 \right]^{\frac{3}{2}}
       } 
+\frac{1}{(2 \pi)} \frac {(s' - s)} {Q^2}
\right\} \lh q \gamma^\mu \;+\ldots \; ,
%\end{aligned}
\end{equation}
and
%Eq.19
\begin{equation}
\mbox{Im} \Pi^\mu(s,s',Q^2)\Big|_{\mbox{HAD}}=F_2(Q^2)\,
\frac{\kappa_p}{2} \left(\frac{s'} {M_N}+
M_N \right)\lh q \gamma^\mu+\ldots
\end{equation}

After substituting the above two spectral functions in the FESR Eq. (15), 
and performing the integrations one obtains
%Eq.20
\begin{equation}
F_2(Q^2)=-\frac {\langle \bar d d\rangle} { 24 \kappa_p \, M_N\, \pi^2\,
 \lambda_N^2}
\left[2 s_0 \left( Q^2 + s_0 \right)  +
    Q^2 \left( Q^2 + 2 s_0 \right) \,
     \ln (\frac{Q^2}{Q^2 + 2 s_0})\right] \; .
\end{equation}

The results for the form factors $F_{1,2}(q^2)$, Eqs. (16) and (20), involve
the free parameters $\lambda_N$ and $s_0$. From QCD sum rules for two-point
functions involving the nucleon current (2) it has been found \cite{QCDSR}-
\cite{MN}
that $\lambda_N \simeq (1 - 3)\times 10^{-2} \mbox{GeV} ^3$, and $\sqrt{s_0}
\simeq (1.1 - 1.5)\, \mbox{GeV}$. The higher values of
$\lambda_N$ and $s_0$ come from Laplace sum rules \cite{MN}, and the lower
values are from a FESR analysis \cite{MNDL} which yields  the relation 
$s_0^3 = 192 \pi^4 \lambda_N^2$. Additional constraints on these parameters
follow, in the present analysis, from the asymptotic behaviour of
$F_1 (q^2)$, and the normalization of $F_2(q^2)$ at $q^2=0$. In fact, from
Eq.(16) it follows that
%Eq.21 
\begin{equation}
   \lim_{Q^2 \rightarrow \infty}
        Q^4 \; F_1(Q^2) = \frac{11 \, s_0^5}{2560 \, \pi^4  \,
        \lambda_N^2} \; ,
\end{equation}        
and from Eq.(20) one finds
%Eq.22
\begin{equation}
F_2(0) \equiv 1 = \frac{- \langle \bar{d} d \rangle \, s_0^2 }
 {12 \, \pi^2  \,\kappa_p \, M_N \, \lambda_N^2} \; .
\end{equation}
Using $\langle \bar{d} d \rangle \simeq - 0.014\, \mbox{GeV}^3$, 
and taking $Q^4 F_1(Q^2) \simeq 1.0 \,\mbox{GeV}^4$ as a 
representative asymptotic value, as indicated by the experimental
data \cite{EXP} in the region
$Q^2 \simeq 10 - 30 \,\mbox{GeV}^2$, the above two equations lead to
$s_0 \simeq 1.2 \, \mbox{GeV}^2$, and $\lambda_N \simeq 10^{-2}
\, \mbox{GeV}^3$, in very good agreement with results from the two-point
function analysis \cite{MN}, \cite{MNDL}. It should be noticed that the
resulting value of $s_0$ is only mildly dependent on the assumed
asymptotic value of the left hand side of Eq. (21), as the determining
equation involves $s_0^3$. Numerically, $s_0$ is well below the Roper
resonance peak, thus justifying the model used for the hadronic spectral
function, Eq.(8).\\ In any case, treating $s_0$ and $\lambda_N$
as free parameters, and performing a least-squares fit to the data on the
proton magnetic form factor $G_M(q^2)$, leads to $s_0 = 1.21 \mbox{
GeV}^2$, and $\lambda_N = 2.1 \times \, 10^{-2} \, \mbox{GeV}^3$,
in line with the values discussed above. The predicted form factor is shown
in Fig.4 (solid line) together with the experimental data \cite{EXP}. The
result from the empirical dipole approximation
%Eq.23
\begin{equation}
G_M^D(Q^2) = \frac{1 + \kappa_p}{(1+\frac{Q^2}{0.71})^2} \; ,
\end{equation}
with $Q^2$ expressed in $\mbox{GeV}^2$, overlaps with the solid line
in Fig. 4 to the extent that the differences cannot be resolved. As may be appreciated
from Fig. 4, the agreement of this QCD sum rule determination with the
data is quite satisfactory.\\
Considering now the neutron form factors, one needs to make the change
$u \leftrightarrow  d$ in Eq.(2). The perturbative QCD spectral function,
Eq.(10), involves now the combination $(\Omega_3 - \Omega_2)$. After
using the FESR Eq.(15) it turns out that $F_1(Q^2)$ for the neutron is
numerically very small and consistent with zero, except near
$Q^2=0$ where it diverges in the chiral limit. In fact, $0 \leq Q^4 F_1(Q^2)
\leq 0.05\; \mbox{GeV}^4$ in the range $0 \leq Q^2 \leq 30 \;\mbox{GeV}^2$.
This smallness of the neutron electric form factor provides a nice self-consistency check
of the method. The Sachs form factors of the neutron, $G_E(Q^2)$ and 
$G_M(Q^2)$, which are then basically determined by $F_2(Q^2)$, turn out
to be almost identical to the dipole fit parametrization for $Q^2\simgt \;
5 \;\mbox{GeV}^2$. This agrees with the experimental data, measured up to
about $Q^2 \simeq 10 \;\mbox{GeV}^2$, although the experimental errors are rather large \cite{GARI}.\\
We comment, in closing, on the next-to-leading order (NLO) contributions
to the three-point function, Eq.(1), which were not considered here. On the
perturbative sector we expect the gluonic corrections to be small, on account
of the extra loop involved, plus the overall factor of $\alpha_s$. The NLO
power correction in the Operator Product Expansion involves the gluon
condensate. This contribution is also expected to be small, as it contains
one more loop with respect to the leading quark condensate term. In addition,
further  suppression of about one order of magnitude would arise from
numerical factors involved in the contraction
of the gluon field tensors. On the hadronic sector, the standard
single-particle pole plus continuum model adopted for the spectral function
is well justified, a posteriori, from the resulting value of the continuum
threshold $s_0$, well below the Roper resonance.

\begin{thebibliography}{99}
\bibitem{REV1} For a review see e.g. V.L. Chernyak,
I.R. Zhitnitsky, Phys. Rep. 112 (1984) 173.
\bibitem{QCDSR} For a recent review see e.g. P. Colangelo, 
A. Khodjamirian, in "{\it At the frontiers of particle physics, Handbook
of QCD}, Vol. 3, 1495, M.A. Shifman, {\it ed.}, (World Scientific,
Singapore, 2001).
\bibitem{MN} For a review see e.g. L.J. Reinders,
H. Rubinstein, S. Yazaki, Phys. Rep. 127 (1985) 1.
\bibitem{MNDL} C.A. Dominguez, M. Loewe, Z. Phys. 
C 58 (1993) 273.
\bibitem{IOFFE81} B.L. Ioffe, Nucl. Phys. B 188 (1981) 817; E:
B 191 (1981) 591.
\bibitem{DLR} C.A. Dominguez, M. Loewe, J.S. Rozowsky,
Phys. Lett. B 335 (1994) 506.
\bibitem{EXP} T. Jansens et al., Phys. Rev. 142 (1965) 922;
W. Bartel et al., Phys. Lett. B 33 (1970) 245; C.H. Berger et al., Phys.
Lett. B 35 (1971) 87; F. Borkowski et al., Nucl. Phys. B 93 (1975) 461;
R.G. Arnold et al., Phys. Rev. Lett. 57 (1986) 174;
R.C. Walker et al., Phys. Lett. B 234 (1989) 353.
\bibitem{GARI} M. Gari, W. Kr\"{u}mpelmann, Z. Phys. A 322 (1985) 689.
\end{thebibliography}

\begin{center}
{\bf Figure Captions}
\end{center}
Figure 1. The three-point function, Eq. (1), to leading order
in perturbative QCD.\\

Figure 2. Triangular and rectangular integration regions of the
Finite Energy Sum Rules, Eq. (15).\\

Figure 3. Non-vanishing terms proportional to the down-quark
condensate, Eq. (17).\\

Figure 4. Experimental data on $G_M^p(Q^2)$ \cite{EXP},
together with the result of this determination (solid line).
The dipole fit, Eq. (23), overlaps almost completely with
the solid line. \\

\newpage
\begin{figure}[tp]
\begin{center}
\epsffile{fig1.eps}
\caption{}
\end{center} 
\end{figure}
\newpage
\begin{figure}[tp]
\begin{center}
\epsffile{fig2.eps}
\caption{}
\end{center} 
\end{figure}
\newpage
\begin{figure}[tp]
\begin{center}
\epsffile{fig3.eps}
\caption{}
\end{center}
\end{figure}
\newpage
\begin{figure}[tp]
\begin{center}
\epsffile{fig4.eps}
\caption{}
\end{center}
\end{figure}

\end{document}



