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\begin{document}

\title{Photoproduction of jets and prompt photons 
at HERA\footnote{\uppercase{T}his work is supported 
by the \uppercase{M}inistry of \uppercase{E}ducation 
of the \uppercase{C}zech \uppercase{R}epublic under the projects
\uppercase{INGO}-\uppercase{LA}116/2000 and 
\uppercase{LN}00\uppercase{A}006.}}

\author{J. CVACH}

\address{Centre for Particle Physics, Institute of Physics AS CR \\
Na Slovance 2, \\ 
Prague 8, CZ-182 21, Czech Republic\\ 
E-mail: cvach@fzu.cz}

\maketitle

\abstracts{
Recent measurements of jet and prompt photon production at HERA are reviewed.
The results are compared to NLO QCD calculations which describe the data well
except in extreme regions of phase space. Studies of the inclusive single jet
cross section provide very accurate value of $\alpha_s$.}

\section{Introduction}\label{sec:intro}
At the positron--proton HERA collider positrons\footnote{We use positron 
for the HERA lepton beam where electrons are used as well.
As the electron data sample has significantly smaller luminosity 
we mention electrons explicitly where it is relevant.} of energy 27.6 GeV
collide with protons with energy 820 (until 1997) or 920 GeV. 
The positrons are the source of quasi-real photons which interact with the
protons. We usually call their interactions
photoproduction at HERA if the photon virtuality $Q^2<1$
GeV$^2$. Scattered positrons at higher virtualities are detected in the main
detector and lead to processes which are called deep inelastic scattering. 
Photoproduction represents the dominant hard scattering process
and provides, therefore, the most accurate results in a large kinematic
range. In the lowest order of the perturbative series expansion in $\alpha_s$,
a quasi-real photon interacts with a parton from the proton directly
(Fig.~\ref{diagr1}a) or as resolved (Fig.~\ref{diagr1}b) via one of its
partons generated in a hadronic fluctuation. Due to the QCD
radiation and fragmentation processes, the partons appear as collimated jets
of particles accompanied in general by the proton remnant and, in the case of
resolved processes, by the photon remnant and by particles resulting from a
possible remnant-remnant interaction, also called the underlying event.
\begin{figure}[ht]
\centerline{\epsfxsize=3.9in\epsfbox{cvach.fig1.eps}}   
\caption{Example LO QCD diagrams for jet-jet photoproduction
in direct (a) and resolved (b) photon interactions. \label{diagr1}}
\end{figure}     
The cross sections for direct and resolved photon interactions are given to LO
by:  
\noindent
\begin{equation*}
\sigma^{ep\rightarrow e\mathrm{jet jet}(\gamma)X}_{\mathrm{dir}} = \sum_j
f_{i/p}\otimes \hat{\sigma}_{ej \rightarrow \mathrm{jet jet}(\gamma)} \,
\end{equation*} 
and for the resolved photon interaction by:
\noindent
\begin{equation*}
\sigma^{ep\rightarrow e\mathrm{jet jet}(\gamma)X}_{\mathrm{res}} = 
\sum_{i,j} f_{\gamma/e} \otimes f_{i/\gamma}\otimes
f_{j/p}\otimes \hat{\sigma}_{ij \rightarrow \mathrm{jet jet}(\gamma)} \,,
\end{equation*} 
where $f_{k/a}$ is the parton density function (PDF)
for parton $k$ in the particle $a=\gamma,p$ and
$f_{\gamma/e}$ is the photon flux in the positron given by the equivalent photon
approximation, $\hat{\sigma}_{ej}$ is the cross section for positron--quark
scattering using the exact matrix element (which is also valid in the NLO case)
and $\hat{\sigma}_{ij}$ is the cross section for parton--parton scattering. PDFs
depend on the factorization scale $\mu_f$ and the cross sections $\hat{\sigma}$ on
the renormalization scale $\mu_r$ via $\alpha_s$. Here both scales are set to be
equal.  

The new HERA results on jet photoproduction are based on
luminosities of several tens of pb$^{-1}$. They allow comparisons to the NLO
calculations up to higher jet $E_T$ and in a more differential way.
Studying high $E_T$ jets has the advantage that the hadronisation of the 
parton from the hard scattering influences the jet less. Also, the
impact of soft processes from the proton--photon remnant interaction are less
important. Having achieved good agreement with the NLO calculations, 
data are used to
check different parametrization of the PDFs of the photon, which are mostly
known from $F_2^\gamma$ measurements at LEP. HERA data are unique due to the
high sensitivity to the gluon density in the photon via the dominant
boson--gluon  fusion process. The sensitivity to the PDFs is high in the region
where the difference between the rapidities
of both jets is small. In contrast, the region with large jet rapidity
difference is sensitive to the form of the matrix element.

We will first present results on single jet inclusive cross section
measurements which provide a very accurate value of $\alpha_s$. Then we review
the results on dijet inclusive cross sections. 
From the four-momenta of both jets, the
fraction of the photon momentum carried by the interacting parton
$x_\gamma^{\mathrm{jet}}$ can be 
estimated. This quantity is used to separate direct and resolved events and to
investigate the dynamics of these classes separately. Finally, new data on
direct photon production are compared to NLO QCD calculations. 

Since most of the results use a similar experimental procedure, we shall
summarize here their main features:
\begin{alphlist}[(d)]
\item $Q^2<1$ GeV$^2$, $0.1<y<0.9$ (H1), $0.2<y<0.85$ (ZEUS).
\item Jets were obtained using the inclusive $k_t$ algorithm\cite{kt}. 
\item Data were corrected for detector effects using two of three MC event
generators, PYTHIA\cite{pythia} and PHOJET\cite{phojet}, which both use LUND
strings for fragmentation and HERWIG\cite{herwig} with its cluster fragmentation
scheme. Soft processes are modeled in PYTHIA as Multiple Interactions
between the proton and photon remnants, PHOJET uses the Dual
Parton Model\cite{dpm} and HERWIG adds an adjustable fraction of the soft
underlying events. 
\item The dominant systematic errors come from two sources: uncertainty in the
knowledge of the energy scale in the hadron calorimeter causes errors of 20\% at
low $E_T$ which 
drop below 10\% at high $E_T$ and model dependence of the detector corrections
which causes errors below 10\%.
\end{alphlist}
For the comparison of data with NLO predictions 
the following calculations are used:
\begin{alphlist}[(d)]
\item NLO weighted parton MC of Frixione\cite{frixio} using GRV-HO\cite{grvho}
for the photon and CTEQ5M\cite{cteq5m} for the proton PDF as the standard
parametrization. To test the dependence of the calculation on the PDF
parametrization, other choices were used such as MRST99 and CVTEQ5HJ for the
proton 
and  AFG-HO for the photon.
\item The program of Klasen, Kleinwort and Kramer\cite{kkk} using the phase
space (PS) slicing method to separate the infrared and mass singular phase
space region. 
\item The program of Fontannaz, Guillet and Heinrich\cite{fgh} for isolated
prompt photon production combines the PS slicing and subtraction methods.
It uses AFG for the photon and MRST2 for the proton PDFs.
\item As the calculations are done for partons, the main uncertainty comes from
the hadronisation correction and decreases from 20\% for low $E_T$ to less than
10\% at high $E_T$. The renormalization and factorization scale uncertainty is
below 10\%.  
\end{alphlist}
Experimental data are displayed in figures with
the inner error bars denoting the statistical error. Systematic errors 
without the uncertainty due to the jet energy scale are added in quadrature with
the statistical error and shown as the outer error bars.
The uncertainty due to the jet energy scale is shown separately as a shaded
band. 

The dominant partonic subprocesses responsible for jet production at HERA are
$\gamma g_p \rightarrow q \bar  q$ and $q_\gamma g_p \rightarrow q g$. The
kinematics of these two-to-two subprocesses are such that the majority of jets
in the region of jet pseudorapidity $\eta^{\mathrm{jet}} < 0$ are predicted to
originate from outgoing quarks while the fraction of gluon-initiated jets
increases as $\eta^{\mathrm{jet}}$ increases\cite{zeus_juc2}. Therefore, the
experimental investigation of jet properties as a function of jet
pseudorapidities allows the study of the characteristics of quark- and
gluon-initiated jets. To distinguish between both types of jets, jet
shapes and mean subjet multiplicities using 
the $k_t$-cluster algorithm were used
as an analyser. The jets become broader 
and the mean subjet multiplicity increases
for final-state gluon-initiated jets. The use of simultaneous cuts on the jet
shapes and subjet multiplicity allowed the definition of enriched samples of
quark- (``thin") and gluon- (``thick") initiated jets.

The distribution of the angle between the jet-jet axis and the beam direction
in the dijet centre-of-mass system $\mathrm{cos}\,\theta^*$ is sensitive to the
spin $J$ of the exchanged particle $d\sigma/d\mathrm{cos}\,\theta^*\sim
(1-|\mathrm{cos}\,\theta^*|)^{-2J}$ as $|\mathrm{cos}\,\theta^*|\rightarrow 1$.
In the
case of direct photon interactions the contributing processes in LO QCD are
$\gamma q(\bar  q) \rightarrow g q (\bar q)$ and $\gamma g \rightarrow q \bar q$
which involve quark exchange in the $s, t$ and $u$ channels. In the case of
resolved photon interactions, the contributing processes $qg \rightarrow qg$,
$q q' \rightarrow q q'$, $gq \rightarrow g q, \ldots$ involve gluon
exchange. The study of the angular distribution of dijet events thus provides a
handle for the investigation of the underlying parton
dynamics\cite{zeus_aachen518}.     

\section{Single jet production and $\alpha_s$}
The new measurements from H1\cite{h1single} cover an extended range of jet
transverse energies 
between 5 and 75 GeV. For $E_T^\mathrm{jet}\ge 21$ GeV, the hadronisation
corrections to the NLO QCD calculation only slightly improve the agreement with
data, whereas for $5 < E_T^\mathrm{jet} < 21$ GeV good agreement can only be
obtained by applying them. The NLO predictions are in
agreement with the $d\sigma/d\eta^\mathrm{jet}$ by measurements in different
$E_T^\mathrm{jet}$ bins except in the lowest range 
$5 < E_T^\mathrm{jet} < 12$ GeV (see Fig.~\ref{d02-225f8}). The data seem to
indicate a rise of the cross section with increasing 
$\eta^\mathrm{jet}$ which is faster than in the theoretical predictions. 
\begin{figure}[ht]
\centerline{\epsfxsize=2.2in\epsfbox{cvach.fig2a.eps}
\epsfxsize=2.25in\epsfbox{cvach.fig2b.eps}}   
\caption{Differential $e^+p$ cross section for inclusive jet production as a
function of (a) $E_T^\mathrm{jet}$ integrated over $-1<\eta^\mathrm{jet}<2.5$.
At the bottom of the figure, the relative difference between data or LO QCD and
the NLO calculations, including hadronisation correction, are shown.
(b) $\eta^\mathrm{jet}$ integrated over the two lowest $E_T^\mathrm{jet}$
ranges. The data are compared with LO and NLO QCD predictions obtained by using
GRV or AFG photon PDFs and CTEQ5M proton PDFs.\label{d02-225f8}}
\end{figure}     
Inadequacy of the photon PDFs in this kinematic range or the absence of higher
order corrections beyond NLO may be responsible.

ZEUS measured\cite{zeussingle} the inclusive jet cross section as a function of
$E_T^\mathrm{jet}$ in the range between 17 and 95 GeV. No significant deviation
with respect to QCD NLO calculation was observed up to the highest scale
studied. The measured cross section $d\sigma/
dE_T^\mathrm{jet}$ was used to determine $\alpha_s(M_Z)$ using the method
presented 
previously\cite{zeus_plb507547}.
\begin{figure}[ht]
\centerline{\epsfxsize=3.2in\epsfbox{cvach.fig3.eps}}   
\caption{(top) The $\alpha_s(M_Z)$  values determined from the QCD fit to the
measured $d\sigma/dE_T^\mathrm{jet}$ values 
in different $E_T^\mathrm{jet}$ regions
(open circles) and the combined value. (lower left) The $\alpha_s(E_T)$ values
determined from the QCD fit of the measured 
$d\sigma/dE_T^\mathrm{jet}$ values as a
function of $E_T^\mathrm{jet}$ (open circles). The solid line represents the
prediction of the renormalization group equation; the light shaded area displays
its uncertainty. (lower right) The $1/\alpha_s(E_T)$ values as a function of
$E_T^\mathrm{jet}$ (open circles). The solid line represents the result of the
two-loop $\alpha_s$ fit to the measured values, the dashed line the
extrapolation of 
the fit to $E_T^\mathrm{jet}=M_Z$. The point plotted at $E_T^\mathrm{jet}=M_Z$
represents the inverse of the combined value from (top). \label{d02-228f4}}
\end{figure}      
The $\alpha_s(M_Z)$ dependence of the predicted $d\sigma/
dE_T^\mathrm{jet}$ in each bin of $E_T^\mathrm{jet}$ was parametrised according
to 
\noindent
\begin{equation*}
[d\sigma/dE_T^\mathrm{jet}(\alpha_s(M_Z))]_i =
C_1^i\alpha_s(M_Z)+C_2^i\alpha_s^2(M_Z) \,
\end{equation*} 
where $C_1^i$ and $C_2^i$ are constants, by using the NLO QCD calculations
corrected for hadronisation effects. Finally, a value of $\alpha_s(M_Z)$ was
determined in each bin of the measured cross section as well as from all data by
a $\chi^2$ fit. The value of $\alpha_s(M_Z)$ is 
\noindent
\begin{equation*}
\alpha_s(M_Z)=0.1224\pm0.0001(\rm{stat.})^{+0.0022}_{-0.0019}(\rm{exp.})
^{+0.0054}_{-0.0042}(\rm{th.})\,.
\end{equation*} 
This value of $\alpha_s(M_Z)$ is consistent with the current world
average\cite{bethke} of $0.1183\pm0.0027$. The largest contribution to the
experimental uncertainty comes from the jet energy scale and amounts to
$\pm$1.5\%.
The QCD prediction for the energy scale dependence of $\alpha_s$ was tested by
determining $\alpha_s$ from $d\sigma/dE_T^\mathrm{jet}$ at different
$E_T^\mathrm{jet}$ values. The measured $\alpha_s(E_T)$ values are shown in
Fig.~\ref{d02-228f4} and are in good agreement with the predicted running of the
strong coupling constant over a large range of $E_T^\mathrm{jet}$.

The cross section $d\sigma/d\eta^\mathrm{jet}$ is presented in
Fig.~\ref{eps03-518f2-3}a for samples of ``thick" and ``thin" jets separated
according to the method explained in Section~\ref{sec:intro}. The measured
cross sections for the two samples of jets exhibit different behaviour: the
$\eta^\mathrm{jet}$ distribution for ``thick" jets peaks at
$\eta^\mathrm{jet}\sim 2$, whereas the distribution from ``thin" jets peaks at
$\eta^\mathrm{jet}\sim 0.7$. The same selection methods has been applied to the
jets of hadrons in the Monte Carlo samples (PYTHIA, HERWIG and PYTHIA MI). The
MC predictions provide a good description of the shape of the ``thin"-jet data
distribution. 
\begin{figure}[ht]
\centerline{\epsfxsize=2.1in\epsfbox{cvach.fig4a.eps}
\epsfxsize=2.1in\epsfbox{cvach.fig4b.eps}}   
\caption{(a) Differential $ep$ cross section $d\sigma/d\eta^\mathrm{jet}$ for
inclusive jet production integrated over $E_T^\mathrm{jet}>17$ GeV for ``thick"
and ``thin" jets. (b) Differential $d\sigma/dE_T^\mathrm{jet}$ cross section
integrated over  $-1<\eta^\mathrm{jet}<2.5$. The PYTHIA prediction normalised to
the total cross section is shown for comparison. \label{eps03-518f2-3}}
\end{figure}     
From the calculation of PYTHIA the sample of ``thick" jets is
predicted to contain 52\% (16\%) of events resulting from $gq(gg)$ subprocesses
in the final state. The 
sample of ``thin" jets contains 65\% (33\%) of events from $qq(qg)$ subprocesses.
The cross 
section $d\sigma/dE_T^\mathrm{jet}$ is shown separately for ``thick" and ``thin"
jet samples.
The ``thin" jet sample exhibits a harder spectrum than the ``thick"-jet
sample. 

\section{Dijet production and photon structure}
Requiring at least two jets per event makes possible the determination of the
distribution of fractional parton momentum $x_p (x_\gamma)$ in the proton (photon)
and of the centre-of-mass scattering angle cos$\,\theta^*$ of the partons
using two-to-two particle scattering kinematics via the expressions:
\begin{equation*}
\begin{array}{lr}
x_p^\mathrm{obs}=\sum^2_{i=1}E_T^{\mathrm{jet}i}
\mathrm{exp}(\eta^{\mathrm{jet}i)}/2E_p,\ \ \ \ & 
x_\gamma^\mathrm{obs}=\sum^2_{i=1}E_T^{\mathrm{jet}i}
\mathrm{exp}(-\eta^{\mathrm{jet}i)}/2E_\gamma, \\ 
\\ 
\mathrm{cos}\,\theta^*=
\mathrm{tanh}(\eta^{\mathrm{jet}1}-\eta^{\mathrm{jet}2})/2 \,.& 
\end{array} 
\end{equation*} 
Both experiments have published new cross sections from photoproduction. The
H1 experiment\cite{h1dijets} concentrated on double differential $x_p$ and
$x_\gamma$ measurements.  The data at the highest $x_p\sim 0.6$ are sensitive to
the gluon 
density of the proton, which contributes almost 40\% of the cross section. The
$d\sigma/dx_\gamma$ cross sections are presented for different regions of
the variables $x_p$ and $E_{T,\mathrm{max}}^\mathrm{jet}(=E_T^\mathrm{jet1}$,
the transverse energy of the highest $E_T$ jet). 
\begin{figure}[ht]
\centerline{\epsfxsize=2.6in\epsfbox{cvach.fig5a.eps}
\epsfxsize=2.1in\epsfbox{cvach.fig5b.eps}}   
\caption{(left) The $x_\gamma$ dependence of the relative difference of the
measured H1 dijet cross sections to the NLO prediction with
hadronisation corrections. 
The two figures show the relative difference for two $E_{T,\mathrm{max}}$
bins. The error due to the uncertainty in the calorimeter energy scale is shown
in the middle. 
The band in the lower plot shows the renormalization and factorization
scale uncertainties. 
(right) Ratio of cross sections from the ZEUS experiment to the NLO prediction
using 
AFG-HO and CTEQ5M1 as the photon and proton PDFs, with the scale set to $E_T/2$.
The theoretical uncertainty is shown as a hatched band.
\label{xgamma}}
\end{figure}
The data (Fig.~\ref{xgamma}~left) are compared to NLO 
calculations with two different parameterizations of the photon structure. The
prediction describes the data well. 
The conclusion of the analysis is that the data
do not require significant changes in the parameterizations of the PDFs but do
provide useful further constraints.

ZEUS published\cite{zeusdijets} dijet cross sections as a function of
$E_T^\mathrm{jet1}$ in different regions of the pseudorapidity 
of the two jets for
$x_\gamma^\mathrm{obs}>0.75$ and $x_\gamma^\mathrm{obs}<0.75$. In general, the
description of the data by NLO predictions is reasonable except when both jets
are produced in the range
$1<\eta^{\mathrm{jet}}<2.4$. Here the NLO prediction lies below the data at low
transverse energy. Similarly to H1, ZEUS tests the current parametrization
of the photon PDFs in different bins of $E_T^\mathrm{jet1}$ (see
Fig.~\ref{xgamma}~right). The predictions using the GRV-HO 
parametrization agree
with the data in the region of lowest transverse energy, but are below the data
for the higher $E_T^\mathrm{jet1}$ bins. The predictions using AGF-HO are
similar in shape to those using GRV-HO but are (10-15)\% lower.

To improve the understanding of the cross section in different
regions of transverse energy, the sensitivity to the value of the transverse
energy cut on the
second jet has been studied. Starting at a minimum of 11 GeV, the cut on
$E_T^\mathrm{jet2}$ was raised in both data and theory for the region
$25<E_T^\mathrm{jet1}<35$ GeV. The data fall less steeply with the increasing
transverse energy than do the NLO QCD predictions and show sensitivity to the
parton densities of the photon. Neither the AFG-HO nor the GRV-HO
parametrization, convoluted with the NLO matrix element, fully describe the
data. The data and NLO calculation cross each other at $E_T^\mathrm{jet2}\sim15$
GeV which is the cut used in the H1 analysis. 
This may explain why in the interval
$25<E_T^\mathrm{jet1}<35$ GeV shown in Fig.~\ref{xgamma}~left, H1 observes good
agreement with the NLO QCD calculation with both photon PDFs, 
as H1 uses a higher cut $E_T^\mathrm{jet2}>15$ GeV than ZEUS.

ZEUS defined dijet data samples of ``thick-thick", ``thin-thin" and
``thick-thin" events according to the criteria mentioned in
Section~\ref{sec:intro}. The differential dijet cross section as a function of
cos$\,\theta^*$ was measured\cite{zeus_aachen518} in the range of
$|\mathrm{cos}\,\theta^*|<0.8$ for dijet invariant mass $M^\mathrm{jj}>52$ GeV. 
The
region of phase space in the $(M^\mathrm{jj},\mathrm{cos}\,\theta^*)$ plane was
chosen in order to minimise the biases introduced by selecting jets with
$E_T^\mathrm{jet1}>17$ GeV and $E_T^\mathrm{jet2}>14$ GeV. 
\begin{figure}[ht]
\centerline{\epsfxsize=2.1in\epsfbox{cvach.fig6a.eps}
\epsfxsize=2.07in\epsfbox{cvach.fig6b.eps}}   
\caption{Differential $ep$ cross section (a) 
$d\sigma/d\mathrm{cos}\,\theta^*_\mathrm{thick}$ for dijet production. The cross
section is for dijet events with 
``thick--thin" selection. The HERWIG prediction is normalised to data at
$\mathrm{cos}\,\theta^*_\mathrm{thick}=0.1$. (b) $d\sigma/dx_\gamma^\mathrm{obs}$
for dijet production with ``thick--thick" (black dots) and ``thin--thin" (open
circles) jet selection. The HERWIG prediction is normalised to the total measured
cross section. Cross sections in (a) and (b) are integrated
over $E_T^\mathrm{jet1}>17$ GeV,  $E_T^\mathrm{jet2}>14$ GeV and
$-1<\eta^{\mathrm{jet}}<2.5$.
\label{eps03-518f5-7}} 
\end{figure} 
Fig.~\ref{eps03-518f5-7}a shows the measured dijet cross section as a
function of $\mathrm{cos}\,\theta^*_\mathrm{thick}$. Calculations using
PYTHIA and HERWIG give a good description of the shape of the measured cross
section. The measured and predicted
cross sections were normalised so as to have a value of unity at
$\mathrm{cos}\,\theta^*_\mathrm{thick}=0.1$. The prediction of HERWIG for the
partonic content is 41\% of $qg$, 6\% of $gg$ and 53\% of $qq$ subprocesses. The
observed asymmetry is adequately reproduced by the calculation and is understood
in terms of the dominance of the resolved 
subprocess $q_\gamma g_p\rightarrow qg$. The
$\mathrm{cos}\,\theta^*_\mathrm{thick}$ distribution is asymmetric due to the
different dominant diagrams in the different kinematic regions.

As the virtuality of the photon increases, the photon
structure gradually changes. Whereas for quasi-real photons, 
the PDF approach is an
indispensable theoretical tool, for $Q^2\gg\Lambda^2_\mathrm{QCD}$ the data can
be analysed using perturbative calculations of the direct
process. However, the resolved photon concept is still useful
phenomenologically provided the photon virtuality remains smaller than a
measure of the hardness of the process, e.g. $E_T^\mathrm{jet}$. 
\begin{figure}[ht]
\centerline{\epsfxsize=3.4in\epsfbox{cvach.fig7.eps}}   
\caption{Triple differential dijet cross
section $d^3\sigma/dQ^2dE_Tdx_\gamma$ for the H1 data depicted by points
compared to the prediction of JETVIP NLO calculations corrected for hadronisation
effects. The dashed line shows the NLO direct cross section, the full line
the sum of NLO direct and NLO resolved contributions. \label{qe3x}}
\end{figure}    
In the new analysis\cite{kamil} 
from H1 of electroproduction of dijets at $2<Q^2<80\,\mathrm{GeV}^2$,
triple differential dijet cross sections were analysed within several
theoretical approaches, differing in the way higher order QCD effects were taken
into account. The NLO direct calculation of JETVIP\cite{jetvip} is shown in
Fig.~\ref{qe3x} by a broken line and is in marked disagreement with the
data in the region of small $Q^2$, small $E_T^\mathrm{jet}$ and small $x_\gamma$.
JETVIP is the only NLO program which calculates also the NLO resolved photon
contribution. The sum of both cross sections direct + resolved, 
is shown by the full
line in Fig.~\ref{qe3x}. The agreement between data and calculations
improves, but is still not satisfactory. The conclusion is that the data show
clear evidence of the need for $\alpha\alpha^3_s$ and 
higher order terms in the theory. The best description of the data 
(not shown here)
is achieved with the LO HERWIG generator with the inclusion 
of longitudinally polarised resolved
photon and initial and final state parton showers as an effective approximation
to higher order QCD processes.

\section{Prompt photon production}
Prompt photons are produced in direct and resolved interactions when a hard
photon emerges instead of a gluon jet. Since the properties of hard photons can
be more accurately measured than those of a jet, prompt photons are expected to
allow cleaner tests of parton dynamics. 
\begin{figure}[ht]
\centerline{\epsfxsize=1.6in\epsfbox{cvach.fig8a.eps}
\epsfxsize=1.6in\epsfbox{cvach.fig8b.eps}
\epsfxsize=1.6in\epsfbox{cvach.fig8c.eps}}   
\caption{Prompt photon differential cross sections as a function of
$E_T^\gamma,\eta^\gamma$ and $\eta^\mathrm{jet}$. The data are compared to LO
(dashed) and NLO calculations. 
The error bands show the effect of a variation of the
renormalization and factorization scales in the NLO calculations from
$0.5E_T^\gamma$ to $2E_T^\gamma$. Dotted line shows the NLO result corrected
for multiple interaction effects using PYTHIA.\label{h1-035}}
\end{figure}    
Two obstacles make these advantages less clear: prompt
photons are born in an environment of copious production of photons 
from $\pi^0$ decays and
their production cross section is a factor of $\alpha/\alpha_s$ 
smaller than that of
gluon jets. H1 has presented new results\cite{h1_aachen093} 
on the photoproduction of
isolated prompt photons, both inclusively and in association with jets. 
The cross sections are given for transverse photon energies and pseudorapidities
of $5<E^\gamma_T<10$ GeV, $-1<\eta^\gamma<0.9$ and for jets in the range
$E_T^\mathrm{jet}>4.5$ GeV, $-1<\eta^\mathrm{jet}<2.3$. The
cross sections for a prompt photon with a jet are presented in Fig.~\ref{h1-035} 
and compared
to NLO calculations\cite{fgh}. The NLO corrections are substantial and lead to
a good description of the data. 
Taking into account the multiple interaction effects
(PYTHIA) improves the data description in particular at $\eta^\gamma>0$ and
$x_\gamma<0.5$. The NLO prediction is not corrected for hadronisation effect. The
H1 cross sections were also compared to the measurements\cite{zeusprompt} of the
ZEUS Collaboration. The results are consistent but the H1 data are somewhat
lower at small $\eta^\gamma$ where the ZEUS data appear to exceed the NLO
calculation.

\section{Conclusions}
The H1 and ZEUS Collaborations have provided accurate measurements of jet and
prompt photon cross sections at the lowest photon virtualities. Data were
compared to several NLO calculations which agree very well at large values of
the
hard scale (usually $E_T^\mathrm{jet}$). The QCD analysis of inclusive jet
$d\sigma/dE_T^\mathrm{jet}$ cross sections leads to accurate value of $\alpha_s$
which are in good agreement with the world average. At the edges of the phase
space of the current measurements (low $E_T^\mathrm{jet}$, large
$\eta^\mathrm{jet}$), especially when cross sections are presented 
in a multi-differential way,
differences between data and NLO calculations become significant. Data provide
constraints on the photon PDF. 

\section*{Acknowledgments}
I am indebted to all members of the H1 and ZEUS collaborations who contributed
to the results. Especially, I would like to thank to my colleagues J. Ch\'yla, 
T. Greenshaw, G. Grindhammer, T. Kluge, R. Lemrani, P. Van Michelen, M. R.
Sutton,  J. Terr\'on and A. Valk\'arov\'a 
who helped me with the preparation of this review. I would like to
thank the organizers of the Ringberg workshop for preparing a stimulating
meeting. 

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