%Paper: 
%From: Stamenov Dimiter <stamenov@ictp.trieste.it>
%Date: Tue, 27 Jun 1995 16:01:20 +0200 (MET DST)


\documentstyle[12pt]{article}
\renewcommand{\baselinestretch} {1.25}
\voffset=-1.5 cm
\hoffset=-12 truemm
\textwidth   157mm
\textheight  23.6cm
%\oddsidemargin 0.0cmhe values of
%\evensidemargin 0.0cmhe values of
%\topmargin 0.0cmhe values of
%\textheight 22.0cmhe values of
%\footheight 1.0cmhe values of
%\headheight 0.0cmhe values of
%\textwidth 16.0cmhe values of

\newcommand{\dst}{\displaystyle}
\def\b{\beta }
\def\a{\alpha }
\def\d{\delta }
\def\L{\Lambda }
\def\ra{\rightarrow }
\def\h{\frac{1}{2}}

\begin{document}

\begin{center}
{\Large\bf Constraints on "Fixed Point" QCD\\
from the CCFR Data on Deep Inelastic\\[2mm]
Neutrino-Nucleon Scattering}
\end{center}
\vskip 2cm


\begin{center}
{\bf Aleksander V. Sidorov
}
\\
{\it Bogoliubov Theoretical Laboratory\\
 Joint Institute for Nuclear Research\\
141980 Dubna, Russia\\
 E-mail: sidorov@thsun1.jinr.dubna.su}
\vskip 0.5cm
\def\thefootnote{*}
{\bf Dimiter B. Stamenov }\footnote{Institute for
Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences\\
Boul. Tsarigradsko chaussee 72, Sofia 1784, Bulgaria.
E-mail:stamenov@bgearn.bitnet }  \\
{\it International Centre for Theoretical Physics, Trieste, Italy}\\
\end{center}

\vskip 0.3cm
\begin{abstract}
The results of LO {\it Fixed point} QCD (FP-QCD) analysis of the CCFR data for
the nucleon structure function $~xF_3(x,Q^2)~$ are presented. The predictions
of FP-QCD, in which $~\alpha_{s}(Q^2)~$ tends to a nonzero coupling constant
$\alpha_{0}$ as $~Q^2\to \infty~,$ are in good agreement with the data.
Constraints for the possible values of the $~\beta~$ function
parameter $~b~$ regulating how fast $~\alpha_ s(Q^2)~$ tends to its
asymptotic
value $~\alpha_{0}~$ are found from the data. The corresponding values of
$~\alpha_{0}~$ are also determined. Having in mind the recent QCD fits to the
same data we conclude that in spite of the high precision and the large
$~(x,Q^2)~$
kinematic range of the CCFR data they cannot discriminate between QCD and
FP-QCD
predictions for $~xF_3(x,Q^2)~$.\\
\end{abstract}
\vskip 0.5 cm
\newpage

{\bf 1. Introduction.}
\vskip 4mm
The success of perturbative Quantum Chromodynamics (QCD) in the description
of the high energy physics of strong interactions is considerable. The QCD
predictions are in good quantitative agreement with a great number of data on
lepton-hadron and hadron-hadron processes in a large kinematic region (e.g. see
reviews \cite{Altarelli} and the references therein).
Despite of this success of QCD,we consider
that it is useful and reasonable to put the question: Do the present data fully
exclude the so-called {\it fixed point}  (FP) theory models \cite{Pol} ? \\

We remind that these models are not asymptotically free. The effective coupling
constant $~\alpha_{s}(Q^{2})~$ approaches a
constant value $~\alpha_{0}\ne 0~$ as $~Q^{2}\to \infty~$
(the so-called fixed point at which the Callan-
Symanzik $\beta$-function $~\beta(\alpha_{0}) = 0~$). Using the assumption
that $~\alpha_{0}~$ is {\it small} one can make predictions for the physical
quantities in the high energy region, as like in QCD, and confront them
to the
experimental data. Such a test of FP theory models has been made  \cite{GR,BS}
by using the data of deep inelastic lepton-nucleon experiments started by
the SLAC-MIT group \cite{SLAC} at the end of the sixties and performed in the
seventies \cite{data70}. It was shown that

$~i$) the predictions of the FP theory models with {\it scalar} and {\it non-
colored (Abelian) vector} gluons {\it do not agree} with the data

$ii$) the data {\it cannot distinguish} between different forms of scaling
violation predicted by QCD and the so-called {\it Fixed point} QCD (FP-QCD), a
theory with {\it colored vector} gluons, in which the effective coupling
constant $~\alpha_{s}(Q^{2})~$ does
not vanish when $Q^{2}$ tends to infinity.\\

We think there are two reasons to discuss again the predictions of FP-QCD.
First
of all, there is evidence from the non-perturbative lattice
calculations \cite{Pat} that the $\beta$- function in QCD vanishes
at a nonzero coupling
$~\alpha_{0}~$ that is small. (Note that the structure of the
$\beta$-function can be studied only by non-perturbative methods.) Secondly,
in the last years the accuracy and the kinematic region of deep inelastic
scattering data became large enough, which makes us hope that discrimination
between QCD and FP-QCD could be performed.\\

Recently we have analyzed the CCFR deep inelastic neutrino-nucleon scattering
data \cite{prep1} in the framework of the {\it Fixed point} QCD.
It was demonstrated \cite{SiSt} that the data for
the nucleon structure function $~xF_3(x,Q^2)~$ are in good agreement with
the LO predictions of this theory model using the assumption that the
{\it fixed point} coupling $~\alpha_{0}~$ is small. In contrast to the results
of the fits to the previous generations of deep inelastic lepton-nucleon
experiments, the value of this constant was determined with a good accuracy:
\begin{equation}
\alpha_{0} = 0.198\pm0.009~.
\label{a0res}
\end{equation}

However, this value of $~\alpha_0~$ is not consistent with $~\alpha_s(M^2_z)~$
measurements at LEP. For instance, from the scaling violation in the
fragmentation
functions in $~e^{+}e^{-}~$ annihilation $~\alpha_s(M^2_z)~$ has been
determined
\cite{fitdel} as:
\begin{equation}
\alpha_s(M^2_z) = 0.118\pm0.005~.
\label{amz}
\end{equation}

This discrepancy follows from the fact that in our analysis pure asymptotic
formula for the structure function $~xF_3~$ has been used , i.e. the effective
coupling constant $~\alpha_s(Q^2)~$ has been approximated with its asymptotic
value $~\alpha_0~$. Results (1) and (2) have shown that in the $~Q^2~$
range studied up to now $~\alpha_0~$ is not yet reached and therefore,
to determine $~\alpha_0~$ properly from the data the preasymptotic behaviour of
$~\alpha_s(Q^2)~$ has to be taken into account.\\

In this paper we present a leading order {\it Fixed point} QCD analysis of the
CCFR data \cite{prep1}, in which the next corrections to the pure asymptotic
expression for $~xF_3(x,Q^2)~$ are taken into account.
We remind that the structure function $~xF_3~$ is a pure non-singlet and the
results of the analysis are independent of the assumption on the shape of
gluons.
As in a previous analysis the method \cite{Kriv} of reconstruction of the
structure
functions from their Mellin moments is used. This method is based on the Jacobi
-
polynomial expansion \cite{Jacobi} of the structure functions.
In \cite{KaSi} this method has
been already applied to the QCD analysis of the CCFR data.\\

{\bf 2. Method and Results of Analysis.}
\vskip 4mm
Let us start with the basic formulas needed for our analysis.

The Mellin moments of the structure function $~xF_3(x,Q^2)~$ are defined as:
\begin{equation}
M_n^{NS}(Q^2)=\int_{0}^{1}dxx^{n-2}xF_{3}(x,Q^2)~,
\label{mom}
\end{equation}
where $~n=2,3,4,...~.$

In the case of FP-QCD  the effective coupling constant $~\alpha_s(Q^2)~$ at
large $~Q^2~$ takes the form:
\begin{equation}
\alpha_s(Q^2) = \alpha_0 + f(Q^2)~,
\label{apre}
\end{equation}
where $~f(Q^2)\rightarrow 0~$ when $~Q^2\rightarrow\infty~.$

Let us assume that $~\alpha_0~$ is a {\it first order} ultraviolet fixed point
for the $\beta$-function, i.e.
\begin{equation}
\beta(\alpha) = -b(\alpha - \alpha_0)~,~~~~~~b~>~0~,
\label{beta}
\end{equation}
Then
\begin{equation}
\alpha_s(Q^2) = \alpha_0 +
[\alpha_s(Q^2_0)-\alpha_0]({Q_0^2\over Q^2})^b~.
\label{aq2}
\end{equation}
and instead of Eq.(5) in \cite{SiSt} we obtain now for the moments
of $~xF_3~$ the following LO expression:
\begin{equation}
M_{n}^{NS}(Q^2)
 =M_{n}^{NS}(Q_{0}^2) \left [ \frac{Q_0^{2}}
 {Q^{2}}    \right ]^{\frac{1}{2}d^{NS}_{n}}F_n(Q^2)~,
\label{mfp}
\end{equation}
where
\begin{equation}
F_n(Q^2) = exp\{{(\alpha_s(Q^2_0)- \alpha_0)\over 2b\alpha_0}
d_n^{NS}[({Q^2_0\over Q^2})^b-1]\}~.
\label{cf}
\end{equation}

In (\ref{mfp}) and (\ref{cf})
\begin{equation}
d_n^{NS} = \frac{\alpha_0}{4\pi}\gamma^{(0)NS}_n
\label{dn}
\end{equation}
and
\begin{equation}
\gamma^{(0)NS}_{n} ={8\over 3}[1 - {2\over n(n+1)} + 4\sum_{j=2}^{n}
{1\over j}]~.
\label{goa0}
\end{equation}

The $n$ dependence of $~\gamma^{(0)NS}_{n}~$ is
exactly the same as in QCD. However, the $~Q^2~$ behaviour of the moments is
different. In contrast to QCD, the Bjorken scaling for the moments of the
structure functions is broken by powers in $~Q^2~$. In (\ref{cf}) and
(\ref{dn})
$~\alpha_0~$ and $b$ are parameters, to be determined from the data.\\

Having in hand the moments (\ref{mfp}) and following the method
\cite{Kriv,Jacobi}, we can write
the structure
function $~xF_3~$ in the form:
\begin{equation}
xF_{3}^{N_{max}}(x,Q^2)=x^{\a}(1-x)^{\beta}\sum_{n=0}^{N_{max}}\Theta_n ^{\a ,
 \beta}
(x)\sum_{j=0}^{n}c_{j}^{(n)}{(\a ,\beta )}
M_{j+2}^{NS} \left ( Q^{2}\right ),   \\
\label{e7}
\end{equation}
where $~\Theta^{\alpha \beta}_{n}(x)~$ is a set of Jacobi polynomials and
$~c^{n}_{j}(\alpha,\beta)~$ are coefficients of the series of
$~\Theta^{\alpha,\beta}_{n}(x)~$ in powers in x:
\begin{equation}
\Theta_{n} ^{\a , \beta}(x)=
\sum_{j=0}^{n}c_{j}^{(n)}{(\a ,\beta )}x^j .
\label{e9}
\end{equation}

$N_{max},~ \alpha~$ and $~\beta~$ have to be chosen so as to
achieve the fastest convergence of the series in the R.H.S.
of Eq.(\ref{e7}) and to
reconstruct $~xF_3~$ with
the accuracy required. Following the results of \cite{Kriv} we use $~\alpha =
 0.12~,
{}~\beta = 2.0~$ and $~N_{max} = 12~$. These numbers guarantee accuracy better
than $~10^{-3}~$.\\

Finally we have to parametrize the structure function $~xF_3~$ at some fixed
value of $~Q^2 = Q^2_{0}~$. We choose $~xF_3(x,Q^2)~$ in the form:
\begin{equation}
xF_{3}(x,Q_0^2)=Ax^{B}(1-x)^{C}~.
\label{e10}
\end{equation}

The parameters A, B and C in Eq. (\ref{e10}) and the FP-QCD
parameters $~\alpha_{0}~$ and $b$
are free parameters which are determined by the fit to the data.

In our analysis the target mass corrections \cite{tmc} are taken into
account.
To avoid the influence of higher--twist effects we have used only the
experimental points in the plane $~(x,Q^2)~$
with $~10 < Q^2\leq 501~(GeV/c)^2~$. This cut corresponds to the following
$~x~$ range:$~0.015\leq x \leq 0.65~$.\\

The results of the fit are presented in Table 1. In all fits only statistical
errors are taken into account.\\

\vskip 0.2 cm
\begin{tabular}{|c|c|c|c|c|c|c|} \hline
$b$&$\chi^2_{d.f.}$& $\alpha_0$ & A & B & C &$\alpha_s(M_z^2)$  \\ \hline
 0.15 & 82.7/61& .057$\pm$.026 & 6.96$\pm$.20 & .799$\pm$.013&3.44$\pm$.03 &
 .121$\pm$.034\\
 0.20 & 82.3/61& .097$\pm$.021 & 6.95$\pm$..20 & .799$\pm$.013 &3.44$\pm$.03&
 .132$\pm$.025\\
 0.25 & 82.0/61& .122$\pm$.018 & 6.94$\pm$.19 & .798$\pm$.013 &3.45$\pm$.03&
 .142$\pm$.020\\ \hline
\end{tabular}
\vskip 4mm
\begin{tabular}{cl}
{\bf Table 1.}& The results of the LO FP-QCD fit
to the CCFR $~xF_3~$ data.\\
&$\chi^2_{d.f.}$ is the
$\chi^2$-parameter normalized to the degree of freedom $d.f.$.
\end{tabular}
\vskip 0.8 cm

Summarizing the results in the Table one can conclude:

1. The values of $~\chi^2_{d.f.}~$  are slightly smaller
than those obtained in the LO QCD analysis \cite{KaSi} of
the CCFR data and indicate a {\it good description} of the data.

2. The values of $~b$, for which the asymptotic coupling $~\alpha_0~$ is
smaller  than $~\alpha_ s( M^2_z)~$, are found to range in the following
interval:
\begin{equation}
0 < b < 0.25~.
\label{ib}
\end{equation}

For the values of $b$ smaller than 0.15 $~\alpha_0~$ can not be determined
from CCFR data. The errors in $~\alpha_0~$ exceed the mean values of this
parameter. For the values of $~b \geq 0.25~$ the mean value
of $~\alpha_0~$ is equal or bigger than $~\alpha_ s(M^2_z)~$.

3. The accuracy of determination of $~\alpha_0~$ is not good enough.
The accuracy increases with increasing $b$.

4. $\alpha_0 = 0.057~$ corresponding to $~b = 0.15~$ is preferred to the
other values of $~\alpha_0~$ determined from the data.

5. The values of $~\alpha_ s(M^2_z)~$ corresponding to the values of $~b~$
from the range (\ref{ib}) are in agreement within one standard deviation
with $~\alpha_ s(M^2_z)~$ determined from the LEP experiments.

6. The values of the parameters A, B and C are in agreement with the
results of \cite{KaSi}. They are found to be independent of $~b~$ and
$~\alpha_0~$. We have found also that multiplying the R.H.S. of (\ref{e10}) by
term $~(1+ \gamma x)~$ one can not improve the fit.\\

{\bf Summary.}
\vskip  4mm
The  CCFR  deep inelastic neutrino-nucleon scattering data have been
analyzed in the
framework of the {\it Fixed point} QCD. It has been demonstrated that the
data for
the nucleon structure function $~xF_3(x,Q^2)~$ are in good agreement with
the LO predictions of this quantum field theory model using the assumption
that $~\alpha_{0}~$
is a {\it first order ultraviolet fixed point} of the $~\beta~$ function and
$~\alpha_{0}~$ is {\it small}.  Some constraints on the behaviour of the
$~\beta~$ function near $~\alpha_{0}~$ have been found from  the data.
The value $~\alpha_{0} = 0.057~$ corresponding to the $~\beta~$ function
parameter $~b=0.15~$ we have obtained is preferred to the other ones
determined from the data.  \\

In conclusion, we find that the CCFR data, the most precise data on deep
inelastic scattering at present, {\it do not eliminate} the FP-QCD and
therefore other tests have to be made in order to distinguish between QCD and
FP-QCD.\\
\vskip 4mm
{{ \bf Acknowledgement}}
\vskip 3mm
We are grateful to Profs. A.L. Kataev, D.I. Kazakov, N.N.
Nikolaev, E.A. Paschos, E. Seiler, D.V. Shirkov and N.G. Stefanis for
useful discussions and remarks. One of us (D.S) would like to thank also the
International Atomic Energy Agency and UNESCO for the hospitality at the
International Centre for Theoretical Physics in Trieste where this work
was completed.\\

This research was partly supported by INTAS (International
Association for the Promotion of Cooperation with Scientists from the
Independent States of the Former Soviet Union) under Contract nb 93-1180, by
the Russian Fond for Fundamental Research Grant N 94-02-03463-a
and by the Bulgarian Science Foundation under Contract \mbox{F 16.}\\

\vskip -1cm

\begin{thebibliography}{99}

\bibitem{Altarelli}
G. Altarelli, in Proc. of  the ``QCD-20 Years Later'' Conf. 9-13 June
1992, Aachen; World Scientific 1993, v. 1., p. 172; Ed. by P. M. Zerwas and
H. A. Kastrup. \\
S. Bethke. Proc. QCD-94 Conference, Montpelier, France, July 1994.

\bibitem{Pol}
A. M. Polyakov,
\newblock {\it ZHETF} {\bf 59} (1970) 542.
G. Mack,
\newblock {\it Nucl. Phys.} {\bf B35} (1971) 592;
A. V. Efremov and I.F. Ginzburg,
\newblock {\it Phys. Lett.} {\bf B36} (1972) 371;
   D. Bailin and A. Love,
\newblock {\it Nucl. Phys.} {\bf B75} (1974) 159.

\bibitem{GR}
M. Gl\"{u}ck and E.Reya,
\newblock {\it Phys. Rev.} {\bf D.16} (1977) 3242; {\it Nucl. Phys.} {\bf B156}
   (1979) 456.

\bibitem{BS}
S. I. Bilenkaya and D. B. Stamenov,
\newblock{\it Sov. J. of Nucl. Phys.} {\bf 31} (1980) 122.

\bibitem{SLAC}
D.H.Coward et al.,
\newblock{\it Phys. Rev. Lett.} {\bf 20} (1968) 292;
E.D. Bloom et al.,
\newblock{\it Phys. Rev. Lett.} {\bf 23} (1969) 930;
H. Breidenbach et al.,
\newblock{\it Phys. Rev. Lett.} {\bf 23} (1969) 935.

\bibitem{data70}
R.G. Roberts and M.R. Whalley,
\newblock{\it J. Phys. G.; Nucl. Part. Phys.} {\bf 17} (1991) D1-D151.

\bibitem{Pat}
\newblock A. Patrascioiu and E. Seiler, {\it Expected Deviations from
Perturbative QCD at 1 TeV or Less}, preprint MPI-Ph/92-18;
{\it Scaling, Asymptotic Scaling and Improved Perturbation Theory,}
preprint MPI-Ph/93-34;
J. Finberg, U. Heller and F. Karsh,
\newblock{\it Nucl. Phys.} {\bf B392} (1993) 493.

\bibitem{prep1}
CCFR Collab., S. R. Mishra et al.,
\newblock Nevis Preprint N 1459 (1992);
CCFR Collab., W. C. Leung et al.,
\newblock {\it Phys. Lett.} {\bf B317} (1993) 655;
CCFR Collab., P. Z. Quintas et al.,
\newblock {\it Phys. Rev. Lett.} {\bf 71} (1993) 1307.

\bibitem{SiSt}
A.V. Sidorov and D. B. Stamenov,
\newblock {\it Preprint JINR} {\bf E2-95-107}, Dubna, 1995.

\bibitem{fitdel}
DELPHI Collab., P. Abrea et al.,
\newblock {\it Phys. Lett.} {\bf B311} (1993) 408.

\bibitem{Kriv}
V. G. Krivokhizhin et al.,
\newblock {\it Z. Phys.} {\bf C36} (1987) 51;
\\V. G. Krivokhizhin et al.,
\newblock {\it Z. Phys.} {\bf C48} (1990) 347.

\bibitem{Jacobi}
G. Parisi and N. Sourlas,
\newblock {\it Nucl. Phys.} {\bf B151} (1979) 421;
\\ I. S. Barker, C. B. Langensiepen and G. Shaw,
\newblock {\it Nucl. Phys.} {\bf B186} (1981) 61.

\bibitem{KaSi}
A. L. Kataev and A.V. Sidorov,
\newblock {\it Phys. Lett.} {\bf B331} (1994) 179.

\bibitem{tmc}
O. Nachtmann,
\newblock {\it Nucl. Phys.} {\bf B63} (1973) 237;
\\S. Wandzura,
\newblock {\it Nucl. Phys.} {\bf B122} (1977) 412.
\end{thebibliography}

\end{document}









