\begin{thebibliography}{99}


%
\bibitem{FFB}
M. Karowski and P. Weisz, Nucl. Phys. {\bf B139} (1978) 455.
%
\bibitem{FFaxioms}
F.A. Smirnov, {\it Form Factors in Completely Integrable
Models of Quantum Field Theory}, World Scientific, 1992.
%
\bibitem{ZZ}
A. B. and Al. B. Zamolodchikov, Ann. Phys. {\bf 120} (1979) 253.
%
\bibitem{Ising}
V. Yurov and Al. B. Zamolodchikov, Int. J. Mod. Phys
{\bf A6} (1991) 3419.\\
O. Babelon and D. Bernard, Phys. Lett. {\bf B288} (1992) 113. 
%
\bibitem{Smirnov}
F. A. Smirnov, Int. J. Mod. Phys. {\bf A9} (1994) 5121.
%
\bibitem{BH} J. Balog and T. Hauer, Phys. Lett. {\bf B337} (1994) 115.
%
\bibitem{BNNPSW} 
J. Balog, M. Niedermaier, F. Niedermayer, A. Patrascioiu, E. Seiler
and P. Weisz, Phys. Rev. {\bf D60} 094508, (1999). 
%
\bibitem{BN} J. Balog and M. Niedermaier, Nucl. Phys. {\bf B500}
(1997) 421.
%
\bibitem{BNII} J. Balog and M. Niedermaier, 
Phys. Rev. Lett. {\bf 78} (1997) 4151.
%
\bibitem{UWolff} U. Wolff, Nucl. Phys. {\bf B334} (1990) 581.
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 and F. Niedermayer, Phys. Lett. {\bf B245} (1990) 522.\\ 
P. Hasenfratz and F. Niedermayer, Phys. Lett. {\bf B245} (1990) 529. 


\end{thebibliography}

Having calculated the square of the 7--particle form factors
the corresponding contributions to the spin--spin correlation function
and the spectral density can now be computed straightforwardly.
The only difficulty is that to get correct numerical results
one has to use high precision arithmetic in order to avoid
huge rounding errors in calculating $G^{(7)}$.
As explained above, to put it into a manageable form, we expressed it
in terms of the basic symmetric polynomials 
$\sigma_1,\dots,\sigma_7$.
Although it is a sum of absolute squares and thus 
it is obviously positive, it is not manifestly positive in this form.
Actually we experienced large cancellations between positive and
negative contributions. After rescaling all rapidities by $\pi$ all
coefficients of the polynomial become integers and we can illustrate
the large degree of cancellation by considering the following
integer rapidities:
\be
\theta_1=12,\quad
\theta_2=11,\quad
\theta_3=10,\quad
\theta_4=9,\quad
\theta_5=8,\quad
\theta_6=7,\quad
\theta_7=0.
\end{equation}
We can exactly calculate the sum of positive and negative
contributions in this case and we get
\bea
{\rm positive\ part}&:&\quad 265129842869261203568
67283491794696305
689824\nonumber\\
{\rm negative\ part}&:&\quad 265129842869261203568
55853186993819121
689824\nonumber\\
{\rm total}&:&\quad \phantom{265129842869261203568}
11430304800877184
000000\nonumber
\end{eqnarray}
showing a 21--digit cancellation here!

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\begin{flushleft}
%\vskip 10mm
\leavevmode
\epsfxsize=140mm
\epsfbox{SPIN7bw.eps}
%\vskip 10mm
\end{flushleft}
\caption{{\footnotesize
Plot of $p^2 I^{\rm spin}(p^2)$ against $\ln(p/M)$, showing a
comparison of the form factor approach to 2--loop perturbation theory.
The solid curve is the FF approximation to $p^2 I^{\rm spin}(p^2)$ obtained by
including all intermediate states with $\le 7$ particles. The lower curves
show the analogous approximations with $\le5,\le3,1$ particles 
respectively. The normalization of the (dashed) PT curve is fixed 
according to (\ref{lambda1}).
Finally the top curve is the PT curve multiplied by a factor $1.05$.
}} 
\label{SPIN7bw}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We have computed the 7--particle contribution to the spin--spin
correlation function using the VEGAS integration routine,
and a subroutine computing $G^{(7)}$ invoking quartic precision (32 digit)
arithmetic. 
The results, compared with the prediction of two--loop perturbation theory,
are shown in Figure~\ref{SPIN7bw}. Note that here we have no free 
parameters
at our disposal in the perturbative calculation as would be the case
in most other models. As discussed above, we know the exact relation between
the perturbative $\Lambda$ parameter and the particle mass $M$ 
and also the absolute normalization of the perturbative curve is
available in this model. The form factor results are in very good
agreement with perturbation theory in the expected (high) energy range.
The small deviation for energies $p/M\sim 10^4$ 
can be accounted for by the contribution of $r>7$ intermediate particles.
To illustrate the fact that this good agreement is quite nontrivial,
in Figure~\ref{SPIN7bw} we also show the perturbative result 
with the overall factor changed arbitrarily by 5\%.

The 7--particle results also corroborate the scaling hypothesis \cite{BN}
for the spectral densities. To study this aspect we introduce the
modified $r$--particle spectral density depending on the logarithmic 
variable $x$ by the definition
\be
\mu\,\rho^{(r)}(\mu)=R^{(r)}(x)\,,\qquad\qquad \mu=Me^x.
\end{equation}
These are defined for all $r\ge2$, where the cases with $r$ even refer
to the spectral densities of the two--point function of the
isospin Noether current. 
The graph of this function is a bell--shaped curve starting as zero
at $x=\ln r$, reaching its maximum $M^{(r)}$ at $x=\xi^{(r)}$ and then slowly
decreasing for larger $x$. Let us introduce the rescaled spectral
density $Y^{(r)}$ by  
\be
Y^{(r)}(z)=\frac{1}{M^{(r)}}\,R^{(r)}\left(\xi^{(r)}z\right).
\end{equation}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[bh]
\begin{flushleft}
%\vskip -10mm
\leavevmode
\epsfxsize=150mm
\epsfysize=90mm
\epsfbox{SSIMCURR7.eps}
%\vskip -4mm
\end{flushleft}
\caption{{\footnotesize
Illustration of the self-similarity property of the rescaled 
spectral densities. The plots show $Y^{(r)}(z)$ (dashed) 
compared with $Y^{(r+1)}(z)$ (solid) for $r=3,4,5,6$.}}
\label{SSIMCURR7}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


It has been found (based on the study of up to 6 particles) that 
the shape of the rescaled spectral density and the parameters
$\xi^{(r)}, M^{(r)}$ satisfy self--similarity 
\be
\lim_{r\to\infty} Y^{(r)}(z)\to Y(z)
\end{equation}
with universal shape function $Y(z)$ and asymptotic scaling,
\be
M^{(r)}\sim M_*\,r^{-\gamma}\,,\qquad\qquad
\xi^{(r)}\sim \xi_*\,r^{1+\alpha}
\end{equation}
for large $r$, with some coefficients $M_*,\xi_*$ and
exponents $\gamma, \alpha$. Whereas the properties of the form factors
are consistent with $\gamma=1$ only, the other exponent can only be 
determined numerically with result $\alpha=0.27$.
For those readers unfamiliar with this model we  
point out the amusing fact that
the self--similarity holds for all $r\ge2$ and thus interrelates
the spectral functions of two different isovector operators,
the spin field and the conserved vector current.
Similarly (25) is equally valid for even and odd $r$ values,
provided we use the normalization introduced in (1).

Self--similarity continues to be satisfied very well as demonstrated
in Figure \ref{SSIMCURR7}. To test asymptotic scaling, we
used the fitted numerical values based on results for up to
6 particles to \lq\lq predict" for the $r=7$ case
\be
M^{(7)}_{{\rm sc}}=0.03188\qquad\qquad{\rm and}\qquad\qquad
\xi^{(7)}_{{\rm sc}}=17.73\,,
\end{equation}
and compared it to the values directly determined from our 7--particle
results
\be
M^{(7)}=0.03189\qquad\qquad\,{\rm and}\qquad\qquad
\xi^{(7)}=17.77.
\end{equation}

Similarly for the integrals
\be
c^{(r)}=\int_M^\infty\,{\rm d}\mu\,\rho^{(r)}(\mu)
=\int_0^\infty\,{\rm d}x\,R^{(r)}(x)\qquad\qquad
h^{(r)}=\int_0^\infty\,\frac{{\rm d}x}{x}\,R^{(r)}(x)
\label{integrals}
\end{equation}
we predict
\be
c^{(7)}_{{\rm sc}}=1.464\qquad\qquad{\rm and}\qquad\qquad
h^{(7)}_{{\rm sc}}=0.04879\,,
\end{equation}
and actually get
\be
c^{(7)}=1.46(1)\qquad\qquad\,{\rm and}\qquad\qquad
h^{(7)}=0.0488(1).
\end{equation}



\documentclass[12pt,a4paper]{article}
\usepackage{amsmath,epsfig}
\setlength{\textwidth}{15cm}
\hoffset=-0.6cm



\begin{document}
 
\setlength{\unitlength}{.8mm}
\input macros


\input title



\section{Introduction}
\input intro


\section{Some basic definitions}
\input definitions


\section{The 7--particle form factors}
\input ff7

\section{The spin--spin correlation function}
\input corr7

\section{Scaling}
\input scaling

\section{Ultra--positivity}
\input ultra

\input ack

%\newpage
\vspace{3cm}

%\appendix
%\section{Appendix}
%\input appen

\input bib

\end{document}







\begin{titlepage}
%
%
\begin{flushright}
%{\bf DRAFT 2}\\
MPI-PhT/03-19\\
April 2003
\end{flushright}
\vspace{1cm}

\begin{center}
{\LARGE \bf Test of asymptotic freedom and scaling hypothesis 
in the 2d O(3) sigma model}
\vspace{2cm}

 
{\large J. Balog$^1$ and P. Weisz$^2$}\\[7mm]

{\small\sl $^1$Research Institute for Particle and Nuclear Physics}\\
{\small\sl H-1525 Budapest 114 Pf. 49, Hungary}
\\[2mm]
{\small\sl $^2$Max-Planck-Institut f\"{u}r Physik}\\
{\small\sl D-80805 Munich, Germany}
\vspace{1cm}

{\bf Abstract}
\end{center}
\vspace{-5mm}

\begin{quote}
The 7--particle form factors of the fundamental spin field of the O(3) 
nonlinear $\sigma$--model are constructed. We calculate the corresponding
contribution to the spin--spin correlation function, and compare 
with predictions from the spectral density scaling hypothesis. 
The resulting approximation to the spin--spin correlation function 
agrees well with that computed in renormalized (asymptotically free) 
perturbation theory in the expected energy range. 
Further we observe simple lower and upper bounds for the sum of the 
absolute square of the form factors which may be of use for 
analytic estimates.

\end{quote}
\vfill
\end{titlepage}
Because of the large cancellation between positive and negative terms
in the representation of $G^{(r)}$ in terms of symmetric polynomials,
it is natural to ask if there exists an alternative representation
that is manifestly positive. It is indeed possible to find such
representations. If we arrange the rapidities in decreasing order
(which is always possible since the polynomial is symmetric under
permutations) then all $u_j=\theta_j-\theta_{j+1}$ rapidity
differences are positive and we found (for all available
particle numbers $3\leq r\leq7$) that if we expand $G^{(r)}$
in terms of these differences then all coefficients are positive.
Moreover, it is possible to find upper and lower bounds both of which
are of the simple form (\ref{perm}) such that, expanded in terms of
the rapidity differences $u_j$, each and every term in the expansion
of $G^{(r)}$ has smaller coefficient than the corresponding one of
the upper bound $P^{(r)}(t^{(r)}_u)$ and similarly larger than the
coefficient of the corresponding term of the lower bound  
$P^{(r)}(t^{(r)}_l)$.

The value of the parameter characterizing the lower bound is
uniformly $t^{(r)}_l=2$ for all cases  $3\leq r\leq7$, whereas
the $t^{(r)}_u$ giving the lowest upper bound are
\be
t^{(3)}_u=2,\quad
t^{(4)}_u=9/4,\quad
t^{(5)}_u=2.434,\quad
t^{(6)}_u=2.576,\quad
t^{(7)}_u=2.688.
\end{equation}
The last three values are numerically approximate 
\footnote{e.g. the exact value for the case $r=5$ is $t^{(5)}_u=208^{1/6}$}
and come from the requirement
that the constant term of $P^{(r)}(t^{(r)}_u)$ be larger than
that of $G^{(r)}$. This is clearly necessary, but at least in the
available cases for $r\ge5$ it is also sufficient
\footnote{The case $r=4$ is an exception since
the requirement in this case yields a value $(34/3)^{1/3}=2.246$,
which is not quite sufficient.}.

The integrals over the spin and current spectral densities are 
given as sums of the integrals defined in (\ref{integrals}):
\begin{eqnarray}
C^{{\rm spin}}&=&\int_M^\infty\,{\rm d}\mu\,\rho^{{\rm spin}}(\mu)
=\frac{4}{\pi}\sum_{k=0}^{\infty}c^{(2k+1)}\,,
\nonumber
\\
C^{{\rm curr}}&=&\int_M^\infty\,{\rm d}\mu\,\rho^{{\rm curr}}(\mu)
=\sum_{k=0}^{\infty}c^{(2k+2)}\,.
\label{infty} 
\end{eqnarray}
An outstanding question is whether these are finite or infinite in 
the FF construction. Certainly renormalized perturbation 
theory predicts that $C^{{\rm spin}}=\infty=C^{{\rm curr}}$.
Also the validity of the scaling hypothesis requires this because 
in this scenario $c^{(r)}$ grows as $\sim r^\alpha$; the numerical 
evidence thereof is shown in Figure~\ref{cr}.

If the bounds we found above for $3\leq r\leq7$ 
continue to be true for all $r$,
then the simple structure of (\ref{perm}) allows to study the structure
of the correlation function analytically. The existence of an upper
limit of simple form may help proving the existence of the correlation
function whereas the existence of the lower bound may facilitate the
construction of a proof that $C^{{\rm spin}}=\infty=C^{{\rm curr}}$
independent of the validity of the scaling hypothesis.


%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\begin{center}
\begin{tabular}[t]{c||c|c|c|c|c|c|c}
$r$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
$c^{(r)}$    & 0.785 & 1.009 & 1.140 & 1.242 & 1.327 & 1.400 & 1.46 \\
$c^{(r)}(2)$ &       &       & 1.140 & 1.206 & 1.229 & 1.225 & 1.200 \\
\hline
\\
$r$ & 8 & 9 & 10 & 11 &  &  &  \\
\hline
$c^{(r)}(2)$ & 1.164 & 1.118 & 1.067 & 1.013 & & &\\
\end{tabular}
\end{center}
\caption{{\footnotesize Numerical values of the integrals 
$c^{(r)}, c^{(r)}(2)$. Numerical errors are estimated as $\pm1$ on the 
last digit quoted.}}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%


We are not going to discuss these questions further in this paper,
but to illustrate the usefulness of the existence of the simple lower bound
we have calculated the integrals $c^{(r)}(2)$. These are defined
analogously to the ones in (\ref{integrals}) using $P^{(r)}(2)$
instead of $G^{(r)}$. The simplicity of the integrand allows us to
calculate these integrals numerically quite effortlessly 
\footnote{This is because one can represent $c^{(r)}(2)$
as an integral with the integrand just involving the polynomial $P_0^{(r)}(2)$
in Eq.~\ref{t} (instead all its permutations), at the cost of extending
the limits of integration over the $u_i$ from $-\infty$ to $+\infty$.}
up to $r=11$. The results of the numerical integration are given in Table~1.

Although the integrals $c^{(r)}(2)$ are (apparently) decreasing with
$r$, it is possible that they are decreasing slow enough to make
the series (\ref{infty}) diverge. Indeed, as seen in Figure~\ref{cr},
$rc^{(r)}(2)$ is monotonically increasing for all $r$ evaluated 
up to now.
\vspace{1cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\begin{flushleft}
\vskip 10mm
\epsfxsize=140mm
\epsfbox{cr.eps}
%\vskip 10mm
\end{flushleft}
\caption{{\footnotesize
Values of $\ln c^{(r)}$ (circles) and of the product 
$rc^{(r)}(2)/(4\pi)$ (squares). The error on $\ln c^{(7)}$ is 
approximately the radius of the circle. 
}}
\label{cr} 
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%\clearpage
