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\begin{document}
\title{Small $x$ PDFs at HERA: Inclusive, Unintegrated, Diffractive}
\author{Victor Lendermann\footnote{~~On behalf of the H1 and ZEUS
Collaborations}}
\institute{Kirchhoff-Institut f\"ur Physik, Universit\"at Heidelberg,
Im Neuenheimer Feld 227, 69120 Heidelberg, Germany}
\maketitle
\begin{abstract}
The present status of HERA measurements of the proton parton distribution
functions (PDFs) in the low $x$ domain is presented. 
PDFs extracted from DIS $ep$ data within the standard factorisation ansatz,
as well as unintegrated PDFs and those describing the diffractive component
of the $ep$ scattering cross section are discussed.
\end{abstract}

\section{Inclusive Analyses}

\subsection{Combination of H1 and ZEUS Data}

Deep inelastic scattering cross sections measured at HERA provide the major
input for the determination of the proton structure at low $x$.
Using the standard QCD factorisation ansatz, the parton distribution functions
are extracted from the doubly differential neutral (NC) and charge (CC) current
cross sections measured as a function of the Bjorken $x$ and
of the four-momentum transfer squared $Q^2$. Over the past two decades,
global fit procedures have been developed which determine the quark and gluon
PDFs of the proton using QCD DGLAP evolution equations
%~\cite{Gribov:1972ri,*Gribov:1972rt,*Lipatov:1974qm,*Dokshitzer:1977sg,*Altarelli:1977zs}
at increasingly higher orders of perturbation theory (see
\cite{Devenish:2004pb} for an overview).
The QCD fits are applied to data sets from a number of different experiments
and consider correlations among the experimental data points.

This traditional extraction procedure however has certain drawbacks in the
treatment of systematic uncertainties.
In particular, correlations through common systematic uncertainties,
both within and across data sets, represent a significant challenge.
The treatment of these correlations is not unique.
In the Hessian method~\cite{Pascaud:1995qs,*Pascaud:2001bi},
each systematic error source is treated as an additional fit parameter,
and the parameters are fitted assuming the model, as provided by
(N)NLO QCD, to optimise the uncertainties and to constrain the PDFs.
In the Offset method (see {\itshape e.g.} \cite{Adloff:1999ah,Chekanov:2005nn})
the data sets are shifted by the effect of each single systematic error
source before fitting, and the resulting fits are then used to form an
envelope function as an estimate of the PDF uncertainty.
All analyses face the problem of data sets not always leading to consistent
results. Some global QCD analyses therefore inflate the PDF uncertainties.
%{\itshape e.g.} $\Delta\chi^2 \gg 1$. The size of the $\chi^2$ variation
%is chosen to ensure that all input data sets are consistent with the
%result of the global fit at a $90$\% confidence level.

The drawbacks mentioned can be significantly reduced by averaging the cross
section data from the different data sets
in a model independent way prior to performing a QCD analysis.
The H1 and ZEUS collaborations presented preliminary results of combining
their HERA\,I data~\cite{H1prelim-07-007,*ZEUS-prel-07-026},
where one averaged value
of the cross section is provided for each measured kinematic point at a given
($x$, $Q^2$, $y$).
Using a method introduced in Ref.\,\cite{Glazov:2005rn}, the correlated
systematic uncertainties are floated coherently allowing each experiment
to calibrate the other. This reduces significantly the
correlated uncertainties for much of the kinematic plane. In addition,
a study of the global $\chi^2$/ndf of the average and of the pull distributions
provides a model independent consistency check between the experiments.

Prior to the combination, the H1 and ZEUS data were transformed to a common
grid of ($x$, $Q^2$) points using ratios of cross sections calculated
based on available PDF parameterisations.
The NC and CC data collected with the proton beam energy of $E_p = 820$\,GeV
were corrected to $920$\,GeV and then combined with the measurements
at $E_p = 920$\,GeV.

As an example, the resulting NC $e^+ p$ cross section data
are shown in Fig.\,\ref{f:h1zeusnc}.
%together with the fits previously
%performed by H1 and ZEUS to their own data..
A precision better than $2$\% is reached in the low $Q^2$ region.
%The correlated systematic uncertainties on each of the combined data points
%make up only a small fraction of the total uncertanty.
%are significantly smaller than the statistical ones. 
%An excellent agreement with the fits is observed.
Comparisons with the fits previously performed by H1 and ZEUS to their own data
have shown an excellent agreement.

\begin{figure}[!tb]
%\centerline{\epsfig{file=H1prelim-07-007.fig4.eps,width=0.8\textwidth}}
\centerline{\epsfig{file=H1prelim-08-045.fig3.eps,width=0.8\textwidth}}
\caption{DIS NC $e^+ p$ scattering cross section from the HERA\,I data taking
period as obtained by combining the published H1 and ZEUS measurements.
%The curves are NLO QCD fits previously performed by H1 and ZEUS
%to their own data.
The predictions of the HERAPDF\,0.1 fit are superimposed.
\label{f:h1zeusnc}}
\end{figure}

%Currently, both collaborations are finishing the analyses of the HERA\,II
%data which should also be included into the combined sample.
At the time of this workshop, H1 presented preliminary results
of the analysis %of the last missing piece
of their HERA\,I $e^+ p$ data
%~\cite{H1prelim-08-043}. The $e^+ p$ data
collected in 1999-2000 in the range $12 \leq Q^2 \leq 150$\,GeV$^2$ and
$2 \cdot 10^{−4} \leq x \leq 0.1$. The data have been combined
with the previously published H1 data in this region using a similar
averaging procedure.
The accuracy of the combined measurement is typically
in the range of $1.5-2$\%.

%\begin{figure}[!tb]
%\centerline{\epsfig{file=H1prelim-08-043.fig7.eps,width=0.75\textwidth}}
%\caption{Electromagnetic proton structure function $F_2$, obtained from
%the combined H1 HERA\,I data, shown as a function of $Q^2$ at various values
%of $x$.
%The data of this measurement (solid, black circles) is complemented
%with the published data at high Q2 (open boxes) and low Q2 (open circles).
%The error bars represent the total measurement uncertainties.
%\label{f:h19900}}
%\end{figure}


\subsection{PDF Fit of the Combined HERA Data}

The H1/ZEUS combined data set has been used as the sole input for a new
NLO DGLAP PDF fit~\cite{H1prelim-08-045,*ZEUS-prel-08-003}.
The consistency of the input data %and its small systematic uncertainties
enables a calculation of the experimental uncertainties of the PDFs using
the $\chi^2$ tolerance, $\Delta \chi^2 = 1$. This represents a significant
advantage compared to the global fit analyses using both HERA and fixed
target data, where increased tolerances $\Delta \chi^2 = 50 - 100$ are used
to account for data inconsistencies. Other advantages of using solely
HERA data are: the absence of heavy target corrections
which must be applied to the $\nu$-Fe and $\mu D$ fixed target data,
and no need to assume isospin symmetry, {\itshape i.e.} that $d$ distribution
in the proton is the same as $u$ distribution in the neutron.

For the new HERAPDF\,0.1 fit, the importance of correlated systematic
uncertainties is no longer crucial, since they are relatively small.
This ensures that similar results are obtained using either Offset or Hessian
method, or by simply combining statistical and systematic uncertainties in
quadrature.

A DGLAP PDF fit analysis depends on a number of model parameters, like
the choice of the starting scale $Q^2_0$ for the evolution, 
the form of the $x$ dependence for PDFs at the starting scale,
the minimum $Q^2$ for the data to fit, $Q^2_{\rm min}$,
the treatment of heavy flavours etc. There are differences in the choices
made by different groups, and in particular, by H1 and ZEUS in their fits
to their own data. In this analysis, both collaborations agreed on a common
set of choices, and variations in the choices were taken to estimate
model-dependent uncertainties (see \cite{H1prelim-08-045,*ZEUS-prel-08-003}
for details).

The predictions of the fit for the NC cross section are superimposed
in Fig.\,\ref{f:h1zeusnc}  on the combined HERA NC data set. The yellow band
shows the total uncertainty including those due to the model dependency.
The total uncertainties of the HERAPDF\,0.1 PDFs are much reduced compared
to the PDFs extracted from the analyses of the separate H1 and ZEUS data sets,
as can be seen in Fig.\,\ref{f:h1zeuspdf}, where the new PDFs are compared
to the ZEUS-JETs and H1PDF2000 PDFs.

\begin{figure}[!tb]
\centerline{%
\epsfig{file=H1prelim-08-045.fig6a.eps,width=0.48\textwidth}%
\epsfig{file=H1prelim-08-045.fig6b.eps,width=0.52\textwidth}}
\caption{Left: PDFs from the ZEUS-JETS and H1PDF2000 fits.
Right: HERAPDF\,0.1 PDFs from the analysis of the combined data set.}
\label{f:h1zeuspdf}
\end{figure}

\subsection{Measurements of $F_L$}

At high inelasticities $y = Q^2 / (xs)$, where $s$ is the $ep$ centre-of-mass
energy squared, the inclusive DIS cross section is sensitive to the size
of the structure function $F_L$ which describes the exchange
of longitudinally polarised bosons. 
In the Quark Parton Model $F_L$ is zero, since due to helicity and angular
momentum conservation a quark with spin
${1 \over 2}$ cannot absorb a longitudinally polarised
photon~\cite{callan-gross}. In QCD,
$F_L$ differs from zero, receiving contributions from quarks and from
gluons~\cite{ZWT,*AM}. 
At low $x$ %in the $Q^2$ region of deep inelastic scattering
(which corresponds to high $y$)
the gluon contribution greatly exceeds the quark contribution.
Therefore $F_L$ is a direct measure of the gluon distribution
to a very good approximation.
An independent measurement of $F_L$ at HERA,
and its comparison with predictions derived from the
gluon distribution extracted from the DGLAP fits,
thus represents a crucial test
on the validity of perturbative QCD at low $x$.
Furthermore, depending on the particular theoretical approach
adopted, whether it be a fixed order pQCD calculation,
a re-summation scheme, or a colour dipole ansatz,
there appear to be significant differences in the predicted
magnitude of $F_L$ at low $Q^2$ mainly due to a large uncertainty
of the gluon PDF.  A measurement of $F_L$
may be able to distinguish between these approaches.

A direct measurement of $F_L$ requires several sets of data taken
at the same $x$ and $Q^2$ but with different $y$ values.
Due to the relationship $y=Q^2/xs$ this requires data to be collected at
different centre-of-mass energies, which was done in the last year
of HERA running, when dedicated runs were performed with lowered proton
beam energies of $E_p = 460$ and $575$\,GeV.

The first HERA measurement of $F_L(x,Q^2)$ was reported by
H1~\cite{Aaron:2008tx}
in the range $12 \leq Q^2 \leq 90$\,GeV$^2$ and $0.0002 \leq x \leq 0.004$.
In this analysis, the scattered electron is reconstructed in the H1
backward calorimeter SpaCal.
Preliminary results were presented by ZEUS in a similar kinematic
range~\cite{ZEUS-prel-08-001}.
Both measurements show a non-zero $F_L$ and are consistent with each other and
with the prediction of (N)NLO QCD fits.
Further preliminary results were presented by H1 in an extended range
of $Q^2$ up to $800$\,GeV$^2$, where the scattered
electron is found either in the SpaCal or in the Liquid Argon calorimeter
covering the central and forward region of the H1
detector~\cite{H1prelim-08-042}.
These results are shown in Fig.\,\ref{f:h1_flav}.

\begin{figure}[!tb]
\mbox{%
\begin{minipage}{0.5\textwidth}
\includegraphics[width=\textwidth]{H1prelim-08-042.fig25_2.eps}%
\end{minipage}\hspace*{0.02\textwidth}%
\begin{minipage}{0.48\textwidth}
\caption{Preliminary results of the H1 measurement of the proton structure
function $F_L$ shown as a function of $Q^2$ at the given values of $x$.
The inner error bars denote the statistical error, the full
error bars include the uncorrelated systematic errors.
%The luminosity uncertainty is not included in the error bars.
The solid curve describes the expectation on $F_L$
from the H1 PDF 2000 fit using NLO QCD.
The dashed (dashed-dotted) curve depicts the expectation of the MSTW (CTEQ)
group using NNLO (NLO) QCD.
The theory curves connect predictions at the
given $(x,Q^2)$ values by linear extrapolation.}
\label{f:h1_flav}
\end{minipage}}
\end{figure}


\section{Unintegrated PDFs}

Using the QCD factorisation theorem, PDFs extracted from DIS data are applied
for the calculation of various scattering processes at hadron colliders, in
particular at the LHC. In practice, the interpretation of experimental data
relies for many signals on analytical calculations performed at a fixed
order of perturbation theory, typically NLO or NNLO (see~\cite{Bern:2008ef} for
a recent review), as well as on Monte Carlo
(MC) event simulations. The major MC programs, PYTHIA~\cite{Sjostrand:2006za}
and HERWIG~\cite{Corcella:2000bw,*Corcella:2002jc}, include
leading order matrix elements for a number of processes, while effects of
higher orders of pQCD are simulated using parton shower models.

For some signatures, %like {\itshape e.g.} multijet production,
especially those with high multiplicity of final state objects,
the complex kinematics %involving multiple hard scales 
and the large phase space available at high energies to be reached at the LHC
make them 
potentially sensitive to effects of QCD initial state radiation arising
from the tail of finite transverse momenta $k_T$ of partonic distributions.
In perturbative fixed-order calculations finite-$k_T$ contributions
are partially accounted for. % through loop corrections.
This is usually
sufficient for inclusive cross sections, but likely not for more exclusive
final state observables. As an illustration, Fig.\,\ref{f:h1dstarjet}\,(left)
from an H1 study of $D^*$+jet photoproduction at HERA~\cite{Aktas:2006ry}
shows the cross section for this process as a function of the difference
in the azimuthal angle $\Delta\phi(D^*,\text{jet})$ between the $D^*$ and
the jet. The lower $\Delta\phi$ tail is significantly underestimated
by the analytical NLO programs FMNR~\cite{Frixione:1994dv,Frixione:1995qc}
and ZMVFNS~\cite{Kniehl:2002xn,Heinrich:2004kj}.

On the other hand, the standard MC programs are based on collinear
evolution of the initial state partons, supplemented by colour
coherence effects for soft gluon emission. It is unknown
whether the approximations involved in these methods will provide
sufficient precision at the LHC energies, as the effects of not
collinearly ordered emissions become increasingly important at low $x$.
A theoretical framework including the finite-$k_T$ contributions
makes use of generalised QCD factorisation technique which involves
PDFs unintegrated not only in the longitudinal but also in the transverse
momenta~\cite{Catani:1990xk,*Catani:1990eg,*Catani:1993ww}
and couples them with suitably defined off-shell matrix elements.
Although MC generators based on this framework~\cite{Jung:2000hk,*Jung:2001hx,Gustafson:2002jy,*Lonnblad:2003wx,*Lonnblad:2004zp,GolecBiernat:2007pu,Hoche:2007hg}
are generally not as developed
as the standard parton shower programs, several studies have demonstrated
their potential advantages over collinear approaches for
specific hadronic final states. This is illustrated in
Fig.\,\ref{f:h1dstarjet}\,(right) in which the same distribution of the
azimuthal angle difference $\Delta\phi(D^*,\text{jet})$ from the H1
study~\cite{Aktas:2006ry} is compared to the prediction of the MC program
CASCADE~\cite{Jung:2000hk,*Jung:2001hx}.
A good agreement with the data is observed in the whole angular range.

\begin{figure}[!tb]
\centerline{%
\begin{minipage}{0.45\textwidth}
\includegraphics[width=\textwidth]{Dstar-jet_DPhi_NLO.eps}%
\end{minipage}\hspace*{0.03\textwidth}%
\begin{minipage}{0.45\textwidth}
\includegraphics[width=\textwidth]{Dstar-jet_DPhi_Cascade.eps}%
\end{minipage}}
\caption{$D^*$+jet cross section as a function of $\Delta\phi(D^*,\text{jet})$
measured by H1 in photoproduction at HERA and 
compared with the prediction of
the next-to-leading order calculations FMNR and ZMVFNS on the left
and of the MC generators PYTHIA and CASCADE on the right.}
\label{f:h1dstarjet}
\end{figure}

Another example is shown in Fig.\,\ref{f:zeus-jets} in which the azimuthal
separation between the two leading jets $\delta\phi$ is plotted
for dijet and three-jet production studied by ZEUS in DIS at
HERA~\cite{Chekanov:2007dx}
and compared to HERWIG and CASCADE predictions~\cite{Hautmann:2008vd}.
CASCADE is superior to HERWIG both in the normalisation and in the shape
of the distribution.

\begin{figure}[!tb]
\centerline{%
\begin{minipage}{0.49\textwidth}
\includegraphics[width=\textwidth]{Dijet_DeltaPhi_0805.1049.eps}%
\end{minipage}\hspace*{0.02\textwidth}%
\begin{minipage}{0.49\textwidth}
\includegraphics[width=\textwidth]{Threejet_DeltaPhi_0805.1049.eps}%
\end{minipage}}
\caption{Dijet (left) and three-jet (right) production cross section
measured by ZEUS in DIS at HERA
as a function of the azimuthal angle $\Delta\phi$ between the two leading
jets in different $x$ intervals and compared to the prediction of
HERWIG and CASCADE.}
\label{f:zeus-jets}
\end{figure}


\section{Diffractive PDFs}

A significant fraction, of the order of $10$\%, of DIS events at HERA
are characterised by a large rapidity gap between hadrons found
in the main detector and the hadronic remnant escaping through the
forward beam pipe. Inclusive diffractive processes are analysed employing
various techniques:
{\itshape (i)} explicitly selecting events with a large rapidity gap;
{\itshape (ii)} extracting the diffractive contribution from a fit
to the invariant mass $M_X$ of the reconstructed hadronic system;
{\itshape (iii)} tagging the scattered proton in the dedicated
forward spectrometers located far away from the main detectors and
very close to the beam pipe (FPS in H1, LPS in ZEUS) and vetoing
the proton dissociation.
The different analyses are based on different statistics and are
characterised by different systematic effects.
All H1 and ZEUS analyses are broadly consistent within the quoted
uncertainties, and the possibility of creating combined H1-ZEUS data sets,
similar to the inclusive HERA data, is currently being considered.

Diffractive events at HERA are successfully described within
the Regge framework~\cite{Collins:1977jy},
in which the rapidity gap is explained by the exchange of a colourless
object lying on the Pomeron trajectory.
The description of the cross section is based on a two-step factorisation
approach. The first step is the standard QCD factorisation, describing
the cross section as a convolution of the matrix element
for the hard scale boson-quark interaction with a PDF in the proton.
The second step describes the PDF as a product of the universal Pomeron flux
in the proton with the diffractive PDF (DPDF).
The Pomeron flux is described by the respective trajectory and depends solely
on the fraction of the proton momentum carried by the Pomeron $x_{I\!P}$ 
and the four-momentum transfer squared at the proton vertex $t$.
The DPDF provides, at a given $Q^2$,
the parton content of the Pomeron for a given longitudinal momentum
fraction $\beta = x/x_{I\!P}$ carried by the struck quark.
Additionally, a small additional term in the second factorisation describes
the Reggeon exchange component.

The second factorisation is an empirical assumption
which is not proven theoretically.
Various experimental studies at HERA have shown this ansatz to work
to a good approximation. However, a recent ZEUS study~\cite{Chekanov:2008cw}
revealed violation of this factorisation, as shown in Fig.\,\ref{f:zeus_fpc}.
Looking in particular at the $x_{I\!P}$ intervals in the central column,
one observes a clear change in the $Q^2$ slope of the structure function
$x_{I\!P} F_2^D(x_{I\!P},\beta,Q^2)$ which is defined similarly
to the conventional structure function $F_2$ in inclusive DIS.
The effect is rather mild, as compared to the typical precision
of the diffractive measurements, and thus should not strongly affect
QCD analyses of diffractive PDFs which are based on this assumption.

The diffractive PDFs, defined in this framework,
were extracted from inclusive diffractive data by H1~\cite{Aktas:2006hy}
in an NLO DGLAP QCD analysis.
%using a five-parameter fit
While the singlet quark distribution is well constrained by the fit,
there is a significant uncertainty of the gluon distribution especially
at high $z_{I\!P}$. Here, $z_{I\!P}$ is the longitudinal
momentum fraction of the parton entering the hard sub-process
with respect to the diffractive exchange, such that $z_{I\!P} = \beta$
for the lowest order quark-parton model process, whereas
$0 < \beta < z_{I\!P}$ for higher order processes.
An additional constraint was obtained from the analysis
of diffractive dijet production in DIS at HERA~\cite{Aktas:2007bv}.
The dijet data which are sensitive to the gluon distribution at high
$z_{I\!P}$ have shown a remarkable consistency with the predictions
from a fit of inclusive diffraction. Including these data into a combined
analyses resulted in a set of the most precise diffractive PDFs currently
available.
Examples of the H1 2007 Jets DPDF fit predictions for the singlet quark
and gluon diffractive PDFs at different factorisation scales $\mu_f^2$
squared, where $\mu_f^2 = Q^2$ in inclusive diffraction,
are shown in Fig.\,\ref{f:h1diffpdf}.

\begin{figure}[!tb]
\mbox{%
\begin{minipage}[b]{0.65\textwidth}
\epsfig{file=ZEUS_desy08-011_fig21.eps,width=\textwidth}%
\end{minipage}\hspace*{0.05\textwidth}%
\begin{minipage}[b]{0.3\textwidth}
\caption{The diffractive structure function of the proton $x_{I\!P}F_2^D(3)$,
as a function of $Q^2$ for different regions of $\beta$ and $x_{I\!P}$,
as measured by ZEUS. The inner error bars show the statistical uncertainties
and the full bars the statistical and systematic uncertainties added in
quadrature. The curves show the result of a
fit to the data
based on a modified BEKW model~\protect\cite{Bartels:1998ea}.
\label{f:zeus_fpc}}
\end{minipage}}
\end{figure}


\begin{figure}[!tb]
\centerline{\begin{tabular}{@{}cc@{\hspace*{0.05\textwidth}}c@{}}
\epsfig{file=H1_DiffPDF_desy07-115_Fig9b.eps,width=0.3\textwidth} &
\epsfig{file=H1_DiffPDF_desy07-115_Fig9c.eps,width=0.3\textwidth} &
\epsfig{file=H1_DiffPDF_desy07-115_Fig9a.eps,width=0.3\textwidth}\\
\epsfig{file=H1_DiffPDF_desy07-115_Fig9d.eps,width=0.3\textwidth} &
\epsfig{file=H1_DiffPDF_desy07-115_Fig9e.eps,width=0.3\textwidth} 
\end{tabular}}
\caption{The diffractive quark (top) and gluon (bottom) PDF
as a function of $z_{I\!P}$ for
two values of the squared factorisation scale $\mu_f^2$: $25$\,GeV$^2$
(left) and $90$\,GeV$^2$ (right). The solid line indicates the H1 2007
Jets DPDF~\protect\cite{Aktas:2007bv},
surrounded by the experimental uncertainty (dark shaded band)
and the experimental and theoretical uncertainties added in quadrature
(ligh shaded band). The dotted and dashed lines show the DPDFs
corresponding to the H1 2006 fits~\protect\cite{Aktas:2006hy}.
\label{f:h1diffpdf}}
\end{figure}



%------------------------------------------------------------------------------
%       Bibliography
%------------------------------------------------------------------------------
\begin{footnotesize}
\bibliographystyle{mpi08} 
{\raggedright
\bibliography{pdf_hera}
}
\parindent0pt
Documents describing HERA preliminary results can be downloaded from the WWW
sites:\\
\begin{tabular}{@{}l@{ }l}
H1:   & https://www-h1.desy.de/publications/H1preliminary.short\_list.html\\
ZEUS: & http://www-zeus.desy.de/public\_results/publicsearch.html
\end{tabular}
\end{footnotesize}
\end{document}

