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%\documentclass[draft]{ws-procs9x6}  
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\begin{document}

%\title{$\alpha_s$ and {\lowercase{$xg$}} from Structure Functions, $F_L$}
\title{Gluon density and Strong Coupling from Structure Functions and
Measurement of $F_L$}

\author{Vladimir Chekelian (Shekelyan)}

\address{Max-Planck-Institut f\"ur Physik \\
Werner-Heisenberg-Institut \\
F\"ohringer Ring 6, 80805 M\"unchen, Germany \\
E-mail: shekeln@mppmu.mpg.de}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% You may repeat \author \address as often as necessary      %
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\maketitle

\abstracts{
In the first phase of data taking at the HERA $ep$ collider
various structure functions were measured. The precision of the
measurements allows to make comprehensive analyses 
in the framework of QCD.
Recent results on the gluon distribution function, the strong coupling
constant $\alpha_s(M_Z^2)\,$ and the longitudinal structure function $F_L$
are reviewed as well as studies of limits of
applicability of the perturbative QCD approach.
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

The study of the structure of hadrons has been a powerful means 
for establishing and testing of the theory of strong interactions
and for the determination of the partonic structure of the nucleon.  
In Quantum Chromodynamics (QCD), structure functions
are defined as a convolution of the universal
parton momentum distributions inside the proton 
and coefficient functions, which contain information
about the exchanged boson-parton interaction. 
At sufficiently large four-momentum transfer squared, $Q^2$,
when the strong coupling $\alpha_s$ is small, 
a perturbative technique
is applicable for QCD calculations of the coefficient 
and splitting functions. The latter
%%%account for parton-parton interactions and 
represent the probability 
of a parton to emit another parton. 
%when scale changes from $Q^2$ to $Q^2$ + $d lnQ^2$. 
Expressions for structure functions are determined by 
convolution integrals of appropriate sums over 
quarks of different flavours and the gluon, and they predict
a logarithmic $Q^2$ dependence (evolution) of the structure functions.
The perturbative QCD (pQCD)  calculations are well established 
within the DGLAP~\cite{dglap} formalism  
to next-to-leading (NLO) order in the strong coupling
and are close to completion to next-to-next-to-leading 
(NNLO) order~\cite{sm}.
 
Precision measurements of structure functions 
in a wide kinematic range allow comprehensive QCD analyses 
to determine the quark and gluon distributions
inside the proton and the strong coupling constant $\alpha_s(M_Z^2)\,$.
Such measurements allow also to explore pQCD application limits, 
for example at low $x$, where the parton
density was found 
to rise steeply towards small $x$ 
by the  H1 and ZEUS experiments at HERA.
Here, $x$ is
the fraction of proton momentum carried by the parton.
In QCD this rise corresponds to an increase of
the gluon density and is expected to slow down at highest energies
(small $x$) due to gluon-gluon recombination, 
the so-called saturation effects. 

Partons, i.e. quarks and gluons, enter differently
into different structure functions. 
In deep inelastic scattering (DIS), 
the cross section of the neutral current (NC) process,
mediated by a $\gamma$ or $Z^{\rm o}$-boson exchange,
can be expressed in terms of three structure functions:
\begin{equation}
  \frac{d^2\sigma^{e^{\pm}p->e^{\pm}X}}{dx dQ^2}
  =\frac{2\pi\alpha^2}{Q^4x}
  \left[Y_+F_2(x,Q^2)-y^2 F_L(x,Q^2)\mp Y_-xF_3(x,Q^2)\right],
\label{dsignc}
\end{equation}
where $y= Q^2/xs$ is the inelasticity,
$s$ is the center of mass energy squared,
$\alpha$ is the fine structure constant
and $Y_{\pm}=1{\pm}(1-y)^2$.
The dominant contribution to the cross section is due to
the proton structure function $F_2$, which is related
in the framework of the quark-parton model (QPM) to the sum of the
proton momentum fractions carried by the quarks and antiquarks
in the proton weighted by the quark charges squared.
The longitudinal structure function $ F_{L}$, 
vanishing in the QPM, is
directly sensitive to the gluon momentum distribution in the proton.
The structure function $ xF_{3}$  is related to valence quarks
and is sizable only at large $Q^2$, 
with $Q^2$ comparable with the $Z^{\rm o}$-boson 
mass squared.

In the first phase of data taking at the HERA ep collider (HERA I),
the  H1 and ZEUS experiments collected about
100 pb{$^{-1}$} of integrated luminosity each in the $e^+p$ mode and 
about 15 pb{$^{-1}$} in the $e^-p$ mode.
These data provided information about structure functions,
which are analysed in the framework of perturbative QCD. 
Further insight into the structure of the proton is obtained 
in semi-inclusive processes such as 
diffraction, heavy flavour and jet production,
%deep-inelastic charm
which are calculable in pQCD.
This report concentrates on recent 
results on the gluon distribution function, the strong coupling
constant $\alpha_s(M_Z^2)\,$ and checks for novel QCD effects.

\begin{figure}[b]
%\epsfxsize=10cm   %width of figure - will enlarge/reduce the figures
%\epsfbox{fig3.eps}
%\figurebox{2cm}{3cm}{} %to have a box alone 
%\centerline{\epsfxsize=3.9in\epsfbox{procs-fig1.eps}}   
%%\centerline{\epsfxsize=2.35in\epsfbox{xglu_h1zeus.diss.eps}
%%\epsfxsize=2.4in\epsfbox{pdf2.eps}}   
\centerline{\epsfxsize=2.35in\epsfbox{chekelian.fig1a.eps}
\epsfxsize=2.4in\epsfbox{chekelian.fig1b.eps}}   
\caption{
Gluon distributions for the dedicated
xg\&$\alpha_s$ fit of H1 and the ZEUS-S PDF fit
for $Q^2$=5, 20, and 200 GeV$^2$ (left).
Parton distributions for the H1 PDF 2000, ZEUS-S PDF and
CTEQ6 fits for $Q^2$=10~GeV{$^2$} (right). 
On the right plot the sea and gluon
distributions are divided by a factor of 20.
\label{gluon}}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{QCD analyses of structure functions}

\begin{figure}[ht]
%\epsfxsize=10cm   %width of figure - will enlarge/reduce the figures
%\epsfbox{fig3.eps}
%\figurebox{2cm}{3cm}{} %to have a box alone 
%\centerline{\epsfxsize=3.9in\epsfbox{procs-fig1.eps}}   
%%\centerline{\epsfxsize=2.3in\epsfbox{ctq6GluUncQ3r.eps}
%%\epsfxsize=2.0in\epsfbox{DESY-02-105_10box.eps}} 
\centerline{\epsfxsize=2.3in\epsfbox{chekelian.fig2a.eps}
\epsfxsize=2.0in\epsfbox{chekelian.fig2b.eps}} 
\caption{
Relative gluon uncertainty as determined in the CTEQ6 fit (left).
Gluon and sea distributions for the ZEUS-S PDF fit (right).
\label{gunc}}
\end{figure}

\begin{figure}[ht]
%\epsfxsize=10cm   %width of figure - will enlarge/reduce the figures
%\epsfbox{fig3.eps}
%\figurebox{2cm}{3cm}{} %to have a box alone 
%%\centerline{\epsfxsize=2.2in\epsfbox{nnlofig30a.ps}
%%\epsfxsize=2.2in\epsfbox{nnlofig40a.ps}}   
\centerline{\epsfxsize=2.2in\epsfbox{chekelian.fig3a.ps}
\epsfxsize=2.2in\epsfbox{chekelian.fig3b.ps}}   
\caption{
Gluon distributions (left) obtained from the LO, NLO and NNLO analyses
of MRST
and corresponding predictions for $F_L$ (right)
as a function of $x$ for various values of $Q^2$.
\label{gflmrst}}
\end{figure}


The HERA I inclusive deep inelastic cross section measurements
at low  $Q^2$ and high $Q^2$
are reviewed in~\cite{dk,bp}. 
The determination of parton distribution functions (PDFs) in QCD analyses
of these data, supplemented by data from 
fixed target experiments, is discussed in~\cite{bp}.
It is interesting to note that available measurements of the NC
and CC (charged current) cross sections in $e^{\pm}p$ interactions
at HERA
allow PDF QCD fits~\cite{h1highq2,zeusqcd},
which lead to a quark flavour decomposition of parton densities
based on HERA data only.

The improved quality of the recent QCD fits
%~\cite{h1lowq2,cteq6,mrst01} 
is achieved by more precise data and 
a new level of sophistication in the fitting techniques including
a full treatment of 
correlated experimental systematic uncertainties.
A new feature of the recent QCD analyses is also
a pragmatic determination of PDF uncertainties
and their physical predictions.
One of the problems of the uncertainty estimation
for fits with many data sets involved is related to
a certain degree of inconsistency among them.
Usually the one sigma error of a parameter in a fit is
determined by variation of $\chi^2$ by one unit from the minimum.
However, 
the level of ``inconsistency'' should
be reflected in the uncertainties.
This can by achieved by the following modification of 
the above criterion:
$\Delta \chi^2 =T^2$~\cite{cteq6,mrst01,durham}, where
$T$ stands for a tolerance which is
estimated from the level of (in)consistency of the data sets
used in a particular QCD fit.
For example, in the CTEQ6 fit the tolerance was taken to be 10
($\Delta \chi^2 =100$).

The H1 collaboration performed also a 
QCD fit~\cite{h1lowq2} devoted especially to the determination
of the gluon distribution and $\alpha_s$ ($xg\&\alpha_s$ fit).
A novel flavour decomposition of $F_2$ used in this fit
allows to reduce the number of
parton distributions to just two combinations 
(apart from the gluon density): 
``valence-like'' and ``sea-like''.
Thus, for this fit there is no need to include deuteron data in the fit,
and the number of data sets can be reduced to a minimum:
the H1 cross section data covering low\,$x$
and the BCDMS $\mu p$ data covering the region of large $x$
(see also section 2.2).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Gluon distribution}

The gluon distributions for different $Q^2$
from the dedicated xg\&$\alpha_s$ fit~\cite{h1lowq2}
of H1, the H1 PDF 2000~\cite{h1highq2}
and the ZEUS-S PDF~\cite{zeusqcd}
 fits are shown in Figure~\ref{gluon}.
The agreement of the different fits
is reasonable, though there are clear differences 
in the shape of the gluon distribution
beyond the quoted error bands.
The CTEQ6 fit, shown in Figure~\ref{gluon} (right), 
bridges the H1 and ZEUS results.
The relative uncertainty band for the gluon distribution
obtained in the CTEQ6 fit is about 10\% at $x<0.3$, 
%with criterion $\Delta \chi^2 =100$
Figure~\ref{gunc} (left).

The differences between the H1 and ZEUS fits
have many sources. The assumptions and procedures
are very different.
This concerns for example
the data sets used in the fits
%%%, treatment of heavy quarks  
and the form of parameterisation of the parton densities
at the starting scale for QCD evolution.
Up to now there is no clear prescription
how to take the last item
properly into account in the uncertainty calculations.
The HERA data themselves look very consistent, 
but it is not excluded that fits are sensitive
to small features of the measurements.
%%%Further investigations are on the way.

The DGLAP evolution leads to a gluon distribution which rises 
towards small $x$ at $Q^2$ above a few GeV{$^2$}, Figure~\ref{gluon}.
The gluon and sea distributions for the ZEUS-S PDF fit at
different $Q^2$, including $Q^2$=1~GeV{$^2$},
are shown in Figure~\ref{gunc} (right).
At the lowest $Q^2$
the rise of the sea density persists, while
the gluon density flattens out at $Q^2$=2.5~GeV{$^2$}
and even becomes negative at $Q^2$=1~GeV{$^2$}.
A simular feature is observed in the MRST NLO fit~\cite{mrst01,mrstnnlo} 
shown in Figure~\ref{gflmrst} (left). While this is not
necessarily a problem in itself, the gluon density
from scaling violations is not an observable,
it results in a distinct unphysical prediction for $F_L$ shown in
Figure~\ref{gflmrst} (right). 
Therefore, $F_L$ measurements at HERA, 
discussed in section 3.3,
are of a particular interest.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Strong coupling constant $\alpha_s(M_Z^2)\,$}

Values for the strong coupling constant
$\alpha_s(M_Z^2)\,$, determined in recent QCD analyses of 
DIS data, are summarised in Table~\ref{tab1}.
The results are rather consistent. 
%except, may be, the GKK result. 
However, the experimental uncertainties of $\alpha_s(M_Z^2)\,$ 
are very different
due to different judgments on the $\Delta \chi^2$ criterion.
The experimental errors are excellent (about 2\%), 
if the canonical criterion $\Delta \chi^2 =1$ is used in the fit.

The criterion $\Delta \chi^2 =1$ can be applied only after careful 
consistency checks of the data sets used in the fit.
For example in the dedicated $xg\&\alpha_s$ H1 fit only data 
from H1 and BCDMS $\mu p$ are involved. 
Furthermore, for the BCDMS $\mu p$ data a cut $y_{\mu} > 0.3$
is applied removing a region 
with large systematics uncertainties, where
the deviations from the QCD fit
(with full treatment of correlated systematics errors) exceed
the systematic errors.
Partial $\chi^2$ of the H1 and BCDMS data points
are shown as a function of $\alpha_s(M_Z^2)\,$ in Figure~\ref{chi2} (right).
The $\chi^2$ minima for both data sets are 
fully consistent and lie around $\alpha_s(M_Z^2)\,=0.115$. 
The importance of the HERA data for the determination of $\alpha_s$
is illustrated in Figure~\ref{chi2} (left),
where $\chi^2$ for fits to ``only H1'' and ``only BCDMS'' 
data are shown. 
Without HERA data the gluon distribution from the fit
to the BCDMS data becomes unrealistically flat at low $x$
and drives $\alpha_s(M_Z^2)\,$ to about 0.110. The HERA data
pin down the gluon density, and, as a result, $\alpha_s(M_Z^2)\,$ moves
to 0.115 if both data sets are used in the fit.

%\begin{table}[ph]
\begin{table}[h]
\tbl{The strong coupling constant from 
NLO and NNLO QCD fits. \vspace*{1pt}}
{\footnotesize
%\tabcolsep7pt
%\begin{tabular}{@{}crl@{}}
\begin{tabular}{|c|r|l|}
\hline
{} &{} &{} \\[-1.5ex]
NLO QCD & $\Delta \chi^2$ & $\alpha_s(M_Z^2)$ \\[1ex]
\hline
{} &{} &{} \\[-1.5ex]
CTEQ6~\cite{cteq6}  &100&$0.1165\pm 0.0065(exp)$\\[1ex]
ZEUS~\cite{zeusqcd}   & $\approx50$
            &$0.1166\pm 0.0008(uncor) \pm 0.0032(cor) \pm 0.0036(norm)$\\[1ex]
         &   &$\pm 0.0018(model) \pm 0.004(theory)$\\[1ex]
MRST01~\cite{mrst01} & 20&$ 0.1190\pm 0.002(exp) \pm 0.003(theory)$ \\[1ex]
H1~\cite{h1lowq2}     &  1&$0.1150\pm 0.0017(exp)^{+~~0.0009}_{-~~0.0005}
~(model) \pm 0.005(theory)$\\[1ex] 
Alekhin~\cite{alekhin}&  1&$ 0.1171\pm 0.0015(exp) \pm 0.0033(theory)$\\[1ex]
%%GKK~\cite{gkk}    & CL&$ 0.112\pm 0.001(exp)$\\[1ex]
\hline
{} &{} &{} \\[-1.5ex]
NNLO QCD & $\Delta \chi^2$ & $\alpha_s(M_Z^2)$ \\[1ex]
\hline
{} &{} &{} \\[-1.5ex]
Alekhin~\cite{alekhin}&  1& $0.1143\pm 0.0014(exp) \pm 0.0013(theory)$ \\[1ex]
\hline
\end{tabular}\label{tab1} }
\vspace*{-13pt}
\end{table}

\begin{figure}[ht]
%\epsfxsize=10cm   %width of figure - will enlarge/reduce the figures
%\epsfbox{fig3.eps}
%\figurebox{2cm}{3cm}{} %to have a box alone 
%\centerline{\epsfxsize=3.9in\epsfbox{procs-fig1.eps}}   
%%\centerline{\epsfxsize=2.3in\epsfbox{fpap22a.eps}
%%\epsfxsize=2.3in\epsfbox{fpap22b.eps}}   
\centerline{\epsfxsize=2.3in\epsfbox{chekelian.fig4a.eps}
\epsfxsize=2.3in\epsfbox{chekelian.fig4b.eps}}   
\caption{
Total $\chi^2$ in the $xg\&\alpha_s$ fits to H1 and BCDMS data separately
and in the common fit to both data sets (left).
Partial $\chi^2$ contributions of the H1 and BCDMS data in the
common fit (right).
\label{chi2}}
\end{figure}

Most of the $\alpha_s(M_Z^2)\,$ determinations in Table~\ref{tab1}
are performed in the NLO QCD approach. A limitation of
the precision of $\alpha_s(M_Z^2)\,$ is due to
the associated theoretical error calculated by varying the
renormalisation and factorisation scales. 
In the NNLO theory this uncertainty is reduced by a factor of 3,
as one can see by comparing Alekhin's NNLO and NLO results,
see also ref.~\cite{sm}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Low-$x$ Regime}

The region of low $x$ accessed at HERA has been the subject
of much debate (e.g. ref.~\cite{Golec-Biernat}). The steep rise of $F_2$ 
discovered already in the first HERA measurements indicates
%manifests 
a new partonic regime governed by a rising gluon density.
%Figure~\ref{gluon}.
%determined from scaling violations Figure~\ref{gluon}
This high parton density regime brings a possibility
to reveal novel QCD effects related to e.g. unitarisation.
The interpretation of experimental results
in the lowest $x$ domain should take into account the
%be taken with care
%%has a complication 
%in view of a 
kinematical correlation between low $x$ and low $Q^2$ at HERA.
The latter domain brings its own complications
related to the transition from the perturbative DIS regime
to the photoproduction limit, where the perturbative
approach is not working due to the large $\alpha_s$
and missing terms in theoretical calculations.
% such as higher twists.
The present experimental situation in
inclusive measurements at low $x$ obtained from detailed studies
of derivatives of $F_2$ in $\ln x$ and $\ln Q^2$ 
and measurements of the longitudinal structure function $F_L$
is discussed below.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Rise of $F_2$ at low $x$}

\begin{figure}[ht]
%\epsfxsize=10cm   %width of figure - will enlarge/reduce the figures
%\epsfbox{fig3.eps}
%\figurebox{2cm}{3cm}{} %to have a box alone 
%%\centerline{\epsfxsize=2.7in\epsfbox{H1prelim-02-041.fig6.eps}
%%\epsfxsize=2.0in\epsfbox{h1lambda.eps}}
\centerline{\epsfxsize=2.7in\epsfbox{chekelian.fig5a.eps}
\epsfxsize=2.0in\epsfbox{chekelian.fig5b.eps}}
\caption{
Local derivatives 
$\lambda = -(\partial \ln F_2(x,Q^2) / \partial \ln x  )_{Q^2}$
(left) and fitted values of $\lambda(Q^2)$ (right).
\label{lambda}}
\end{figure}

In the double asymptotic limit, the DGLAP evolution equation
can be solved and $F_2$ is expected to rise approximately as a power
of $x$ towards low $x$. A power behaviour is also predicted
in BFKL theory~\cite{bfkl}. A damping of this rise
would indicate the presence of novel QCD effects. 
A relevant observable
for the investigation of the dynamics of this growth is
the partial derivative of $F_2$ w.r.t. $\ln x$ at fixed $Q^2$  
\begin{equation}
 \lambda = -(\partial \ln F_2(x,Q^2) / \partial \ln x  )_{Q^2}~.
\label{lamdef}
\end{equation}

The high precision of the present $F_2$ data allowed H1 to measure
this observable locally~\cite{h1rise,h1prel}, as it is shown 
in Figure~\ref{lambda} (left) for the low $Q^2$ region.
The measurements are consistent with no dependence
of $\lambda$ on $x$ for $x<0.01$. 
Thus, the monotonic rise of  $F_2$ persists
down to the lowest $x$ measured at HERA, and 
no evidence for a change of this behaviour such as 
a damping of the growth is found.
%We can state that no damping effects of the rise of $F_2$
%are visible yet at present energies for $Q^2 > 0.85$~GeV$^2$.

The observed independence of the local derivatives in $\ln x$
at fixed $Q^2$ suggests that
$F_2$ can be parameterised in a very simple form
\begin{equation}
 F_2 = c(Q^2) x^{-\lambda(Q^2)}~.
\label{f2par}
\end{equation}
The results for $\lambda(Q^2)$ are shown in Figure \ref{lambda}
(right). The coefficient $c(Q^2) \approx 0.18$
and  the parameterisation $\lambda(Q^2) = a \cdot $ln($Q^2/\Lambda^2$)
for $Q^2 \geq 2$~GeV$^2$
are in accord with pQCD predictions. 
In contrast, at $Q^2 \leq 1$~GeV$^2$ the behaviour is 
changing, and, in the low $Q^2$ limit, $\lambda$
is approaching 0.08,
which is expected from
%which corresponds to
the energy dependence of soft hadronic interactions
$\sigma_{tot} \sim s^{\alpha_P(0)-1} \approx s^{0.08}$.

\begin{figure}[ht]
%\epsfxsize=10cm   %width of figure - will enlarge/reduce the figures
%\epsfbox{fig3.eps}
%\figurebox{2cm}{3cm}{} %to have a box alone 
%%\centerline{\epsfxsize=2.3in\epsfbox{d00-181f12_col.eps}
%%\epsfxsize=2.5in\epsfbox{f2slope.eps}}
\centerline{\epsfxsize=2.3in\epsfbox{chekelian.fig6a.eps}
\epsfxsize=2.5in\epsfbox{chekelian.fig6b.eps}}
\caption{
The derivative $(\partial F_{2} / \partial \ln  Q^{2})_x\,$
as function of x for different $Q^2$.
\label{deriv}}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Scaling violations of $F_2$ }

Another important quantity in view of
possible non-linear gluon interaction effects is
the derivative 
\begin{equation}
 (\partial F_{2}(x,Q^2) / \partial \ln  Q^{2})_x\,
\label{f2deriv}
\end{equation}
%%$(\partial F_{2} / \partial \ln  Q^{2})_x\,$
describing scaling violations.
Its behaviour in $x$ 
is a direct reflection of the gluon density dynamics
in the associated kinematic range.

The derivatives measured by H1 and ZEUS are shown in Figure~\ref{deriv}
as a function of $x$ for different $Q^2$.
They show a continuous growth towards low $x$
down to lowest $Q^2 = 0.3$ GeV$^2$, without an indication
of a change in the dynamics.
The derivatives are well described by the pQCD calculations 
for $Q^2 \geq 3$ GeV$^2$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Longitudinal structure function $F_L(x,Q^2)$}

\begin{figure}[ht]
%\epsfxsize=10cm   %width of figure - will enlarge/reduce the figures
%\epsfbox{fig3.eps}
%\figurebox{2cm}{3cm}{} %to have a box alone 
%\centerline{\epsfxsize=3.9in\epsfbox{procs-fig1.eps}}   
%%\centerline{\epsfxsize=3.05in\epsfbox{H1prelim-03-043.fig10.eps}   
%%\epsfxsize=2.05in\epsfbox{03-004-04.ps}}
\centerline{\epsfxsize=3.05in\epsfbox{chekelian.fig7a.eps}   
\epsfxsize=2.05in\epsfbox{chekelian.fig7b.ps}}
\caption{
Summary of the $F_L$ measurements by H1 
at fixed photon-proton center of mass energy 
$W=276$~GeV, $W \approx \sqrt{sy}$  (left).
The $F_L$ measurement by ZEUS using radiative events (right).  
\label{fl}}
\end{figure}

Non-zero values of the structure function $F_L$ 
appear in pQCD due to gluon radiation. 
Thus, $F_L$ is the most
appropriate quantity to test QCD to NLO and especially to examine
the pathological effects 
of the negative gluon distribution 
discussed in section 2.1.


According to eq.~\ref{dsignc}, the $F_L$ contribution to
the inclusive cross section is significant only at high $y$. 
%%%due to a factor of $y^2$ in front of $F_L$.
The conventional way to measure $F_L$ is to explore 
the $y$  dependence of the cross section at given
$x$ and  $Q^2$ by changing the center of mass energy of the
interaction. Such a measurement is not yet performed at HERA.
The H1 collaboration developed other ways to determine $F_L$ 
from the measured turn over of the cross section at high $y$. 
This becomes possible due to the H1 measurements
at very low scattered electron energy
down to 3 GeV. The determinations of $F_L$ are based on
various methods to propagate $F_2$ measurements to
the region corresponding to high $y$ (low $x$).

A summary of the $F_L$ measurements by H1~\cite{h1fl}
is shown in Figure~\ref{fl} (left).
The results are significantly above zero everywhere,
including the lowest $Q^2$.
They are compared with pQCD predictions 
and different phenomenological models 
showing that already at the present level of precision
the measurements can distinguish 
different calculations.

The ZEUS collaboration used DIS radiative events
with detection of the photon from the initial state radiation (ISR) 
process in a zero angle photon tagger. ISR  
effectively reduces the energy of the electron
participating in the interaction.
The result~\cite{zeusfl} is shown in Figure~\ref{fl} (right). 
Large errors of the measurement
suggest that precise direct measurements of $F_L$ at HERA
can be performed only by reducing the beam
energy and employing the highest $y$ domain.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\section{Conclusions}

The inclusive DIS cross sections from HERA I
cover a kinematic phase space extended 
towards low $x$ and high $Q^2$ by two orders of magnitude
compared with fixed target experiments.
These data, studied in the framework of perturbative
QCD in the form of NLO DGLAP analyses, 
provide an accurate determination
of the gluon distribution function. The strong coupling constant
$\alpha_s(M_Z^2)\,$ determined in such analyses has small
experimental errors 
and is currently limited in precision by 
the accuracy of the NLO theory. 
No evidence for new QCD effects like saturation
has been found in inclusive processes at low $x$.

The HERA II program, aiming for a 10-fold increase of statistics
with a  longitudinally polarised electron beam,
with much improved H1 and ZEUS detectors,
is just starting.
A special run with lower proton beam energies 
is envisaged for direct measurements of $F_L$ at low $x$.
%is part of the program.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Acknowledgments}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end{thebibliography}

\end{document}





