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\begin{document}

\title{
\hfill{\large INP Report 1666/PH (1994)}\\
\vspace{1 cm}
       {\bf Gluons from logarithmic slopes of $F_{2}$
            in the NLL  approximation}
\vspace{1cm}
\author{
       K.Golec--Biernat \thanks{e--mail address:  golec@bron.ifj.edu.pl}
\\[15pt]
{\it Department of Theoretical Physics}
\\
{\it Institute of Nuclear Physics}
\\
{\it Radzikowskiego 152, Krak\'ow, Poland}
\\
       }
\date{}
      }


\maketitle
\vspace{3cm}
%------------ABSTRACT---------------------------
\begin{abstract}
 We make a critical, next--to--leading order, study of the accuracy
 of the "Prytz" relation, which is frequently used to extract
 the gluon distribution at small $x$ from the logarithmic slopes of
 the structure function $F_{2}$. We find that the simple relation
 is not genarally valid in the HERA regime, but show that it is a reasonable
 approximation for gluons which are sufficiently singular at small $x$.
\end{abstract}

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%------------MAIN BODY--------------------------

\newpage
 Recently a new method of determination of the gluon distribution in
 the nucleon from the deep-inelastic scattering (DIS) data was proposed
  \cite{prytz}.
 The method is based on an approximate relation between the gluon
 density and the logarithmic slopes of the nucleon structure function $F_{2}$,
 derived using the leading logarithmic approximation (LLA) formulae.
 The crucial point in the derivation is the observation that for the
 existing parametrizations of the parton distributions
 the logarithmic slope of $F_{2}$ at small $x$ ($x < 10^{-2}$)
 depends mostly on the gluon density and the sea quarks
 contribution can be neglected. In this case
 the following formula for the
 logarithmic slope of $F_{2}$ is valid
 (for $n_{f}=4$)
 \be
   \label{slope1}
   \frac{dF_{2}}{d logQ^{2}} \approx \frac{5 \alpha_{s}}{9 \pi}
     \int_{x}^{1} dz\ G(\frac{x}{z},Q^{2})\ P_{qg}(z)\ ,
 \ee
  where $P_{qg}(z) = (1-z)^{2}+z^{2}$ is the Altarelli--Parisi splitting
  function in LLA.

  The integral in (\ref{slope1}) was approximated
  by the value of the gluon distribution at the point $2 x$,
  and the final relation was found
  \be
    \label{slope3}
    \frac{dF_{2}}{d log Q^{2}}(x)\approx \frac{10 \alpha_{s}}{27 \pi}\ G(2x)\ .
  \ee
  to be valid at small $x$ below $10^{-2}$.

  Relation  (\ref{slope3}) was  used by the H1 colaboration at DESY
  to estimate the gluon
  distribution in the proton from the first measurement of the structure
  function $F_2$ in the DIS region at the e-p
  colider HERA at DESY\cite{h1}.
  The slope at l.h.s of relation (\ref{slope3}) was determined at
  $Q^{2} = 20~GeV^{2}$ from the data and the gluon distribution was found
  to exhibit strong  rise when $x \rightarrow 0$.

  Relation (\ref{slope3}) helps to estimate the gluon distribution
  in LLA, while most of existing predictions of the small $x$ behaviour
  of the parton distributions are done in the next-to-leading logarithmic
  approximation (NLLA).
  It means that the corrections which are
  one order higher in  $\alpha_{s}$ are taken into account both in
  the Altarelli--Parisi equations
  and in the structure functions
  to determine  the parton distributions.
  A natural question arises whether relation (\ref{slope3}) remains
  a good approximation also for NLLA gluon ditributions.
  The author of paper \cite{prytz} claims that this relation  is
  valid with an accuracy of around $20~\%$, taking into account many
  uncertainties, also those resulting from  higher order  $\alpha_{s}$
  corrections.
  We will show that this is not generally true in NLLA.
  The validity of relation (\ref{slope3})
  crucially depends on a type of gluon distribution.

  In order to show this we computed the logarithmic slope of $F_{2}$
  on l.h.s of formula (\ref{slope3}) for three different sets of existing
  parton distribution parametrizations at small $x$ ; MRS parton distributions
  $D^{'}_{-}$ and $D^{'}_{-}$ \cite{mrs} , and GRV ones \cite{grv}.
  All of them are extrapolations from the region of high $x$, where
  the fixed targed DIS data exist.
  These distributions represent the whole spectrum of possible behaviours
  of the sea quarks and gluon distributions at small $x$.
  The Lipatov motivated $D^{'}_{-}$ parametrization
  predicts strong rise of gluon and sea, proportional to $x^{-0.5}$ when
  $x \rightarrow 0$, whereas $D^{'}_{0}$ distributions are flat. They
  tend to constant values in the limit of small $x$.
  The gluon and sea distributions from the GRV set lie in between. They
  are much steeper than $D^{'}_{0}$ distributions but not as steep as
  $D^{'}_{-}$ ones.
  We computed the slope of $F_{2}$
  numerically, using a part of a computer program,
  which solves the Altarelli--Parisi equations in NLLA, prepared to analyse
  the DIS data from HERA \cite{gol}.
  Having done that we compared the computed exact slopes of $F_{2}$ to
  r.h.s of relation (\ref{slope3}), which  can easily be found for the parton
  parametrizations under consideration.

  The results of this analysis are shown in {\bf Fig.1}. We plot  there
  the exact and approximate slopes (r.h.s of relation (\ref{slope3}))
  computed at $Q^{2} = 20~GeV^{2}$,
  as well as their ratios.
  It is easy to see that the  validity of
  relation (\ref{slope3}) in NLLA depends strongly on the parton distributions.
  While for the steep gluons from the $D{'}_{-}$ set the relation is still
  approximately true, then for the flat type $D^{'}_{0}$ gluons
  it is dramatically violated. For the GRV parametrization relation
  (\ref{slope3}) could be acceptable but for $x$ bigger then $10^{-4}$.

   In order to clarify the results from {\bf Fig.1} we computed
   contributions of different terms to the exact NLLA formula for the
   logarithmic slope of $F_{2}$.
   This formula  has a form of
   (\ref{slope1}), where the NLLA splitting function $P^{(1)}_{qg}$ \cite{fur}
   was added to LLA one
   \be
   P_{qg}(z) \rightarrow P_{qg}(z) + \frac{\alpha_{s}}{2 \pi} P^{(1)}_{qg}(z)
   \ee
   In addition,
   there are also terms with quark distributions
    with LLA and NLLA contributions in the exact formula for the slope.
   In  {\bf Fig.2} we plotted
   all four contributions to $d F_{2}/d logQ^{2}$ at $Q^{2} = 20~GeV^{2}$,
   coming
   from the gluon ({\it on the left}) and quark ({\it on the right})
   terms computed in LL  ({\it dashed lines}) and NLL ({\it dotted lines})
   approximations.
   The exact slopes ({\it continuous lines}) are also shown.
   The computation was performed for the  parton distribution sets under
   consideration. We immediately
   see that the quark contributions in both approximations
   can be neglected in the derivation of the approximate relation. The slope
   of $F_{2}$ is determined mostly by the gluon terms. For the singular
   MRS($D^{'}_{-}$) gluons the LLA contribution dominates, while for the flat
   MRS($D^{'}_{0}$) ones the NLLA contibution is as important as the LLA
   one. This is the reason why the approximate relation (\ref{slope3}) is
   not valid in NLLA for the flat type gluon distributions. In such case
   relation (\ref{slope3}) applied to extract gluon from data may overestimate
   the steepness of the gluon distribution.
   We illustrate the last statement in {\bf Fig.3},
   where the exact gluons used to compute the  slope of $F_{2}$
   and the gluons estimated from relation (\ref{slope3}) are plotted.

  In conclusion, the simple relation (\ref{slope3})
  between the logarithmic slope of $F_{2}$ and the gluon distribution
  is not generally
  valid in the NLL approximation. It remains a reasonable approximation
  only for gluons which are sufficently singular at small $x$.

  Fortunately, the gluons estimated by the H1 collaboration from the first
  data seem to be  steep enough to exclude the case with flat-like  gluons.
  Nevertheless, the problem of determination of the gluon distribution
  at small $x$ is much deeper from the theoretical point of view.
  We have discussed this problem within the standard approach based on the
  Altarelli--Parisi evolution equations and the standard factorization
  formula. A different, more appropriate at small $x$ approach was proposed
  \cite{cat,ask}. It is based on the Lipatov equation and
  a new factorization theorem.
   This approach may lead in principle to different results from
   those based on the Altarelli--Parisi equations \cite{slp}.
   A future analysis of  new data from HERA should clarify these problems.

\vskip 1cm

\noindent{\Large\bf Acknowledgment}

  I would like to thank J. Feltesse,  M.W. Krasny, J. Kwieci\'nski,
  A.D.Martin and J.Turnau  for helpful comments and discussions.
  The EC "Go  West" fellowship is gratefully acknowledged.
  This work has been partially supported by the Polish Committee
  for Scientific Research under the grant 2 0198 91 01.
\newpage
%----------REFERENCES-----------------
\begin{thebibliography}{111}
\bibitem{prytz} K.Prytz, Phys. Lett.\ {\bf 311B} (1983) 286.
\bibitem{h1} H1 Collab., I.Abt. et al.,
              Phys.Lett. {\bf B321} (1994) 161,
\bibitem{mrs} A.D.Martin, W.J.Stirling and R.G.Roberts,
              Phys.Lett.\ {\bf 306B} (1993) 145.
\bibitem{grv} M.Gluck, E.Reya and A.Vogt: DO-TH 91/07: see also Z. Phys.
                  {\bf C48} (1990) 471.
\bibitem{gol} K.Golec--Biernat, W.Krasny, S.Riess,\ in preparation.

\bibitem{fur} W.Furma\'nski and R.Petronzio, Phys.Lett. {\bf B97} (1980) 437,
                Z. Phys. {\bf C11} (1982) 263.

\bibitem{cat} S.Catani, F.Fiorani and G.Marchesini, Phys.Lett.\ {\bf 234B}
               (1990) 339;\ Nucl.Phys.{\bf B357} (1991) 167,\
               S.Catani, M.Ciafaloni and F.Hautman Phys.Lett. {\bf B242}
                (1990) 97;\ Nucl.Phys. {\bf B366} (1991) 135.
\bibitem{ask} A.J.Askew, J. Kwieci\'nski, A.D.Martin and P.Sutton,\
               Phys.Rev. {\bf D47} (1993) 3775,\
               Mod. Phys. Lett. {\bf A8 } (1993) 3813,\
               Durham preprint DTP/93/28 to appear in Phys.Rev.D.

\bibitem{slp} A.J.Askew, K.Golec-Biernat, J. Kwieci\'nski,
              A.D.Martin and P.Sutton,\
              IFJ preprint 1653/PH, Krakow (1993) to appear in Phys.Lett. B

\end{thebibliography}




\newpage

\noindent{\large {\bf Figure Captions}}

\begin{itemize}

\item[Fig.\ 1]
      Exact slopes computed in NLLA (continuous lines)
      versus slopes resulting from the
      approximate Prytz relation assumed to be still valid in NLLA
      (dashed lines). The slopes are computed for
      MRS($D^{'}_{-}$), MRS($D^{'}_{0}$) and GRV parton ditributions.
      On the right hand side the ratio of the approximate to exact
      slopes is shown. All quantities are computed at $Q^{2} = 20~GeV^{2}$.

\item[Fig.\ 2]
      Parton contributions to the exact slope $dF_{2}/d logQ^{2}$ (continuous
      lines)
      at $Q^{2} = 20~GeV^{2}$ computed  for parton distributions under
      consideration in NLLA.
      Contibution from gluons is
      shown on the left and from quarks on the right.
      Dashed and  dotted
      lines show  the leading log  and
      next--to--leading  contribution respectively .


\item[Fig.\ 3]

      Comparison of the exact gluons used to compute the exact slopes of
      $F_{2}$ in NLLA (continuous line)
      to those estimated from the approximate Prytz relation
      (dashed line) at $Q^{2} = 20~GeV^{2}$. The ratios of the approximate
       gluon distributions to the exact ones are shown on the right hand side.

\end{itemize}

\end{document}

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 2422 l 67 2417 l 68 2414 l 71 2410 l 76 2409 l 79 2409 l 84 2410 l 87 2414 l
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 2417 l 89 2422 l 87 2425 l 84 2428 l s 56 2441 m 89 2441 l s 56 2441 m 56 2461
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49
 2533 m 52 2536 l 57 2539 l 64 2542 l 71 2544 l 78 2544 l 86 2542 l 92 2539 l
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87
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2645
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2641
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2663
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89
 2703 l 89 2709 l 87 2712 l 84 2715 l 81 2717 l 76 2718 l 68 2718 l 64 2717 l
60
 2715 l 57 2712 l 56 2709 l 56 2703 l cl s 83 2707 m 92 2717 l s 51 2726 m 50
 2726 l 49 2727 l 48 2728 l 47 2730 l 47 2733 l 48 2734 l 49 2735 l 50 2736 l
52
 2736 l 54 2735 l 56 2734 l 64 2726 l 64 2737 l s 2 lw 751 778 1250 1961 bl 1
lw
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l
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1420
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1413
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1802
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 1782 1101 l s 1795 1134 m 1795 1101 l s 1795 1134 m 1809 1134 l 1814 1133 l
 1816 1131 l 1817 1128 l 1817 1125 l 1816 1122 l 1814 1120 l 1809 1118 l 1795
 1118 l s 1806 1118 m 1817 1101 l s 1849 1130 m 1846 1133 l 1841 1134 l 1835
 1134 l 1830 1133 l 1827 1130 l 1827 1126 l 1828 1123 l 1830 1122 l 1833 1120 l
 1843 1117 l 1846 1115 l 1847 1114 l 1849 1111 l 1849 1106 l 1846 1103 l 1841
 1101 l 1835 1101 l 1830 1103 l 1827 1106 l s 1871 1141 m 1868 1138 l 1865 1133
 l 1862 1126 l 1860 1118 l 1860 1112 l 1862 1104 l 1865 1098 l 1868 1093 l 1871
 1090 l s 1882 1134 m 1882 1101 l s 1882 1134 m 1894 1134 l 1898 1133 l 1901
 1130 l 1903 1126 l 1905 1122 l 1905 1114 l 1903 1109 l 1901 1106 l 1898 1103 l
 1894 1101 l 1882 1101 l s 1917 1109 m 1914 1109 l 1913 1106 l 1912 1102 l 1912
 1100 l 1913 1096 l 1914 1093 l 1917 1093 l 1918 1093 l 1921 1093 l 1922 1096 l
 1923 1100 l 1923 1102 l 1922 1106 l 1921 1109 l 1918 1109 l 1917 1109 l cl s
 1930 1141 m 1933 1138 l 1936 1133 l 1940 1126 l 1941 1118 l 1941 1112 l 1940
 1104 l 1936 1098 l 1933 1093 l 1930 1090 l s 1863 296 m 1863 263 l s 1882 285
m
 1878 283 l 1875 280 l 1874 275 l 1874 272 l 1875 267 l 1878 264 l 1882 263 l
 1886 263 l 1890 264 l 1893 267 l 1894 272 l 1894 275 l 1893 280 l 1890 283 l
 1886 285 l 1882 285 l cl s 1923 285 m 1923 259 l 1921 255 l 1920 253 l 1917
252
 l 1912 252 l 1909 253 l s 1923 280 m 1920 283 l 1917 285 l 1912 285 l 1909 283
 l 1905 280 l 1904 275 l 1904 272 l 1905 267 l 1909 264 l 1912 263 l 1917 263 l
 1920 264 l 1923 267 l s 1947 302 m 1944 299 l 1940 294 l 1937 288 l 1936 280 l
 1936 274 l 1937 266 l 1940 259 l 1944 255 l 1947 252 l s 1956 285 m 1974 263 l
 s 1974 285 m 1956 263 l s 1983 302 m 1986 299 l 1990 294 l 1993 288 l 1994 280
 l 1994 274 l 1993 266 l 1990 259 l 1986 255 l 1983 252 l s gr
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