STRUCTURE OF VIRTUAL PHOTONS AT HERA~\footnote{T\lowercase{alk given at} PHOTON 2001 C\lowercase{onference}, A\lowercase{scona}, S\lowercase{witzerland}, S\lowercase{eptember} 2001}
K. SEDL\'AK~\footnote{S\lowercase{upported by} GA AV\v{C}R \lowercase{grant} B1010005
\lowercase{and projects} INGO-LA 116/2000, LN00A006}
Triple differential dijet cross-sections in $e^\pm p$ interactions measured with the H1 and ZEUS detectors at HERA are presented. % in the region %of low photon virtuality $2~{\GeV}^2 < Q^2 < 80~{\GeV}^2$ and low transverse jet %momentum $E_t^{jet} > 5~\GeV$. The data are compared to Monte Carlo simulations which differ in their assumptions about photon structure and parton evolution. Effects %connected with of the resolved processes of longitudinally polarized virtual photons at HERA are investigated for the first time.
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\newcommand{\pb}{\rm pb}
\newcommand{\cm}{\rm cm}
\newcommand{\xg}{$x_\gamma$\,}
\newcommand{\Et}{$\overline{E}_t^2$\,}
\newcommand{\qq}{$Q^2$\,}
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\begin{document}

\title{STRUCTURE OF VIRTUAL PHOTONS AT HERA~\footnote{T\lowercase{alk 
      given at} PHOTON 2001 C\lowercase{onference}, A\lowercase{scona},
    S\lowercase{witzerland}, S\lowercase{eptember} 2001}}

\author{K. SEDL\'AK~\footnote{S\lowercase{upported by} 
       GA AV\v{C}R \lowercase{grant} B1010005, 
       \lowercase{and projects} INGO-LA 116/2000,
       LN00A006}}

\address{Institute of Physics, AS CR, Na Slovance 2,
Praha 8, 182 21, Czech Republic\\E-mail: ksedlak@fzu.cz\\{\rm On 
behalf of the H1 and ZEUS Collaborations}}

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\maketitle

\abstracts{
Triple differential dijet cross-sections in $e^\pm p$ interactions
measured with the H1 and ZEUS detectors at HERA are presented.
% in the region
%of low photon virtuality $2~{\GeV}^2 < Q^2 < 80~{\GeV}^2$ and low transverse jet
%momentum $E_t^{jet} > 5~\GeV$. 
The data are compared to Monte Carlo
simulations which differ in their assumptions about photon structure
and parton evolution. Effects 
%connected with 
of the resolved
processes of longitudinally polarized virtual photons at HERA
are investigated for the first time.}

%================================================================
\section{Dijet Production at HERA}
\label{uvod}
The production of dijet events at HERA is dominated by processes,
in which a virtual photon, radiated from 
the electron, interacts with a parton in the proton.
%f the photon virtuality is high enough, we can
%escribe such processes by a perturbatively calculable
%CD hard interaction of virtual photon with a parton
%mitted from the proton only, usually called ``direct'' interaction.
In the region of photon virtuality $Q^2 \gg \Lambda_{\mathrm QCD}^2$, hard collisions
   of the photons do not necessitate the introduction of the concept
   of the resolved photon (as for the real photon) and the process can in
   principle be described by the direct photon contribution alone, in which 
   the photon interacts as a whole with partons from the proton.

%The aim of this analysis is to study an intermediate region 
%of low $Q^2$, %$2~\GeV < Q^2 < 80 GeV$, 
%where at least three different
%theoretical approaches can be employed:

%The aim of this analysis is to study 
The analyses presented here explore the region 
   $\Lambda_{\mathrm QCD}^2 \ll Q^2$ \raisebox{-0.8ex}{$\stackrel{\mathrm{<}}{\sim}$} 
%{\stackrel{<}{$\sim$}} 
$E_t^2$, where different theoretical approaches can
   be used to take into account higher order corrections.

%\begin{enumerate}
%\item
\vspace{0.8em}
{\bf a) LO direct and resolved interactions} based on the DGLAP 
     evolution equations and parton showers allows
%represent 
%     a conventional theoretical approach to  
%     describe our measurement. In a previous  
%     analysis of H1 experiment~\cite{Tana}, the effective virtual photon
%     PDF has been determined within this scheme. 
%
%     One of the aims of analyses in the DGLAP approach is 
     the effects of transversally and
%, for the first time at HERA, 
     also
     longitudinally polarized resolved photon 
     interactions to be studied~\cite{chyla,Chyla:2000hp}.
%     for the first time at HERA.
     Since the cross section for longitudinal photons vanishes 
     in the photoproduction
     regime due to gauge invariance, and 
%on the other hand
     the concept of a resolved photon breaks down for $Q^2 > E_t^2$,
%     at high photon virtualities,
     the only evidence for this phenomena can be
     observed in the region $0 < Q^2 \ll E_t^2$.
     The main difference between the longitudinal ($\gamma^*_L$)
     and transverse ($\gamma^*_T$) virtual photon
     arises from the dependence of the respective fluxes on $y$. 
     While for $y\rightarrow 0$,
     both transverse and longitudinal fluxes are approximately same,
     the longitudinal flux vanishes for $y\rightarrow 1$.
     Also the dependence of the point-like (i.e. perturbatively calculable)
     part of the photon Parton Distribution Functions (PDF) 
     on \qq and $E_t^2$
     differs -- while the $\gamma^*_T$ PDF 
     is proportional to $\ln (E_t^2/Q^2)$, the
     $\gamma^*_L$ PDF depends on the scale in a typically hadron-like 
     manner.
%     PDF should not depend neither on $E_t^2$, nor on \qq.


\vspace{0.8em}
%\item
{\bf b) $\mathbf k_t$ unordered initial QCD cascades} accompanying the hard 
     process, employed for example in BFKL or CCFM evolution, 
     can lead to final 
     states in which the partons with the
     largest $k_t$ may come from the cascade, and not, as in DGLAP
     evolution, from the hard subprocess. Such 
     events may have a similar topology as is observed for the resolved 
     interactions introduced in the previous paragraph.
     This possibility is investigated using the CASCADE 
     generator~\cite{Cascade,Cascade2}
     based on the CCFM evolution equation.

\vspace{0.8em}
%\item
{\bf c) NLO calculations} with and without
%either pure or even extended by 
     the concept
     of the resolved virtual photon offer another natural 
     possibility to be examined. Such comparisons are envisaged.
%not yet ready,
%     however analyses presented here enable to do so directly. 
%\end{enumerate} 

\vspace{0.8em}
All of these three approaches include higher order corrections
to the LO QCD matrix elements; however, each of the approaches treats them
differently.
%, i.e. sums up different parts of higher order corrections.

%================================================================
\section{Measurement of Dijet Cross-Section}

The selection criteria defining the H1 and ZEUS dijet samples are
summarized in Table~\ref{tabulka}:
%
%
\begin{table}[ht]
\begin{center}
\begin{tabular}{|l|l|}
\hline 
{\bf H1} ~~ (16.3 pb$^{-1}$, 1999) & {\bf ZEUS} ~~ (38.2 pb$^{-1}$, 1996-97)\\ \hline 
$\sqrt{s}=318~\GeV$ & $\sqrt{s}=300~\GeV$ \\ \hline
$2~{\GeV}^2 < Q^2 < 80~{\GeV}^2$ & $0.1 < Q^2 < 10\,000~{\GeV}^2$ \\ \hline
$0.1 < y < 0.85$ & $0.2 < y < 0.55$ \\ \hline
%$k_t$ long. invariant jet algorithm, $\gamma^* P$ CMS.
$E_t^{jet\,1,2} > 5~\GeV$   &   $E_t^{jet\,1} > 7.5~\GeV$ \\
$\overline{E}_t > 6~\GeV$   &    $E_t^{jet\,2} > 6.5~\GeV$ \\ \hline
$-2.5 < \eta^{jet\,1,2} < 0$  &   $-3 < \eta^{jet\,1,2} < 0$ \\ \hline
\end{tabular}
%\vspace{3ex}
%\begin{tabular}{|l|l|}
%\hline 
%{\bf H1} & { \bf ZEUS} \\ \hline
%
% { HERWIG 5.9, RAPGAP 2.8:} & { HERWIG 5.9:} \\
%  ~~~   proton PDF: GRV LO & ~~~ proton PDF: MRSA\\
%  ~~~   $\gamma^*_T$ PDF: SaS1D & ~~~ $\gamma^*_T$ PDF: SaS1D\\
%  ~~~  $\gamma^*_L$ PDF: J. Ch\'yla & \\
%  ~~~  s.u.e. in HERWIG: 10\% &  ~~~ s.u.e. in HERWIG: 0\% \\ \hline
%
%{ CASCADE 1.0:} & \\ 
%    ~~~   Default setting & \\ \hline
%\end{tabular}
%
\caption{Selection criteria of the dijet samples.}
\label{tabulka}
\end{center}
\end{table}
%
%
$Q^2$ denotes the photon virtuality, $y$ is the inelasticity,
$s$ is the total electron-proton centre-of-mass energy, 
$E_t^{\rm jet\,1,2}$ and
$\eta^{\rm jet\,1,2}$ are
the transverse energy and pseudorapidity of the jet with the highest 
or second highest
$E_t$, and $\overline{E_t}$ is defined as 
$(E_t^{\rm jet\,1}+E_t^{\rm jet\,2})/2$.

The measured data are corrected for detector effects using a bin-to-bin
method (ZEUS) or Bayesian unfolding (H1).  The largest source of
systematic
errors arises from the main calorimeter calibration uncertainty and, in
the case of H1, also from a model dependence of the detector correction.
%procedure.
The ZEUS measurement was presented in more detail at the
EPS 2001~\cite{Zeus}.

\begin{figure}[t]
\begin{center}
\epsfig{file=zeus1.ps,width=27pc,height=35pc,%
bbllx=45pt,bblly=229pt,bburx=526pt,bbury=710pt,clip=}
%\caption{Triple differential}
%\end{figure}
%\begin{figure}[t]
%hep \vskip 1.cm
%hep \epsfig{file=zeus4.ps.gz,width=17pc,height=23pc,%
%hep bbllx=75pt,bblly=218pt,bburx=520pt,bbury=655pt,angle=270,clip=}
%hep \begin{picture}(1,1)
%hep \put(-300,0){\Large a)}
%hep \put(-300,-180){\Large b)}
%hep \end{picture}
\caption{Triple differential dijet cross-section 
$d^3\sigma_{ep}/dQ^2 d\overline{E}_t^2 dx_\gamma$ for the ZEUS data
is depicted by points, the shaded bands display an uncertainty estimate
arising from the main calorimeter calibration.
%%%%(points) is compared to the HERWIG MC prediction (histograms).
The dashed histograms represent the direct part of HERWIG only, 
the full line stands for the sum of the direct and transverse resolved
contributions of HERWIG.}
\label{Zeus1}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\epsfig{file=zeus4.ps,width=27pc,%
bbllx=75pt,bblly=218pt,bburx=520pt,bbury=655pt,clip=}
\caption{The ratio of cross-sections
$R=\sigma(x_\gamma < 0.75) / \sigma(x_\gamma > 0.75)$ from 
Fig.~\ref{Zeus1}.}
\label{Zeus2}
\end{center}
\end{figure}

%================================================================
\section{Results and Discussion}

The corrected triple-differential dijet cross-section measured
by ZEUS as a function of \qq, \Et and \xg is shown in
Fig.~\ref{Zeus1}.
A prediction of HERWIG with the SaS1D parameterization of 
%transversally polarized photon 
the \gammaT
PDF, as well as the direct contribution of HERWIG
is compared to the data. Since the overall normalization of 
the LO Monte Carlo simulation is to some extent uncertain, the HERWIG
prediction has been normalized to the highest \xg bin $(x_\gamma > 0.75)$
in the data. The normalization is done separately for each 
$(Q^2,\EtEQ)$ bin.

In the region where $Q^2 > \EtEQ$, the data are well described by the
direct HERWIG component only.  
%On the other hand, 
Resolved interactions are needed if $Q^2 < \EtEQ$.
%in order to get the HERWIG prediction closer to the measurement.
However, even with the \gammaT resolved processes included,
HERWIG tends to underestimate 
the data in the lowest \qq region for $x_\gamma < 0.75$. 

This fact is also demonstrated in Fig.~\ref{Zeus2} by the ratio
of $\sigma(x_\gamma < 0.75) / \sigma(x_\gamma > 0.75)$. % is plotted.
The slope of this distribution can be interpreted as %showing the 
a suppression of the virtual photon structure with increasing photon
virtuality.


The corrected triple-differential dijet cross-section measured
at H1 as a function of \qq, \Et and \xg is shown in Fig.~\ref{H1}.
%In contrast to Fig.~\ref{Zeus1}, 
The H1 measurement is performed
in a different phase space (see Table~\ref{tabulka}) and Monte Carlo
predictions are not normalized to the data.

A comparison of the H1 measurement with HERWIG
leads to similar conclusions as 
%discussed 
drawn above
for ZEUS.  In addition, we can see that for the
highest \qq range ($25 < Q^2 < 80 ~\GeV^2$) and $x_\gamma<0.75$, 
the HERWIG direct contribution almost describes the data 
in the lowest \Et bin, but is too low in the highest \Et bin.  
This indicates an importance of the resolved processes
even at high \qq, once the hard scale, \Et, is high enough.

Standard HERWIG with direct and \gammaT resolved
contributions underestimates the data. The description 
%could be
is improved by adding \gammaL  resolved photon 
interactions,
% This possibility is investigated using 
which is done using a slightly
modified HERWIG with the correct longitudinal photon flux and 
a recent $\gamma^*_L$ PDF parameterization~\cite{Chyla:2000hp}.
As demonstrated in Fig.~\ref{H1}, the \gammaL resolved
contribution is significant, and brings  HERWIG
closer to the measurement.

%On the other hand, one may ask to what extent could the simple 
%   enhancement of PDF of \gamma_T^* lead to similar ...

On the other hand, a simple enhancement of the PDF of the $\gamma^*_T$ 
in the 
%transverse 
resolved 
%HERWIG 
contribution could lead to a similar prediction
as the introduction of $\gamma^*_L$.
% Due to a large experimental error
% of the data, $\gamma^*_T$ and $\gamma^*_L$
%
%For this reason, 
To eliminate this ambiguity,
%we also studied 
the dijet cross-section has also been studied
as a function of \qq, \xg\, and $y$, which is shown in Fig.~\ref{H2}.
HERWIG is below the data, even if the resolved \gammaL
is added. 
This may be due to the uncertainty of the overall normalization of the
LO Monte Carlo prediction. 
In the region where $x_\gamma<0.75$, the slope of the HERWIG prediction 
depends significantly on whether $\gamma^*_L$ processes are included or
not. The $\gamma^*_L$ contributes significantly at low $y$,
while it becomes very small compared to $\gamma^*_T$ at high $y$. 
Unlike a pure enhancement
of $\gamma^*_T$ resolved processes by a constant factor,
the addition of $\gamma^*_L$ brings
the $y$ dependence of HERWIG much closer to the measurement.
%, while pure enhancement
%of $\gamma^*_T$ by a constant factor leads to a significantly worse
%description of the $y$ dependence.


As motivated in Section~\ref{uvod},
the measured cross-sections in Fig.~\ref{H1} are also compared 
to a prediction of the CASCADE MC program based on the CCFM evolution
approach. 
%If we keep in mind that 
This theoretical concept does not involve 
any information about the virtual photon structure and involves
many fewer free parameters for tuning than the usual DGLAP-like MC
programs. Nevertheless, CASCADE describes the data well, 
except for the \qq dependence.
The \qq behavior, however, is related to the parameterization of
the unintegrated PDFs %gluon density functions 
used in the program, which are not yet
constrained unambiguously.

%\figurebox{20pc}{15pc}{} % to have a box alone
%\epsfxsize=10pc % will enlarge or reduce the postscript figures based on the xsize
%\rotatebox{90}{
%\epsfxrot=90
%\epsfbox{H1prelim-01-133.fig6.eps} % postscript image file name

\begin{center}
%\begin{flushleft}
%\begin{minipage}[t]{3in}
\begin{figure}[ht]
%\rotatebox{90}{
%
\epsfig{file=H1prelim-01-133.fig6.eps,width=27.7pc,angle=0}
%\caption{Triple differential dijet cross-section as 
%  a function of \xg, \Et and $Q^2$. The data are
%  compared to HERWIG and RAPGAP MC predictions. 
%  Separately shown are the direct (dir) and the resolved
%  component of transversally polarized $\gamma$  (res$_T$).}
%\label{fig6}
%}
%\end{figure}
%\end{minipage}


%\begin{figure}[t]
%\epsfxsize=10pc
\caption{Triple differential dijet cross-section 
$d^3\sigma_{ep}/dQ^2 d\overline{E}_t^2 dx_\gamma$ for the H1 data
depicted by points is compared to predictions of the HERWIG and
CASCADE MC programs.\hspace{2cm}}
\label{H1}
\end{figure}
\end{center}
\begin{center}
\begin{figure}
\epsfig{file=H1prelim-01-133.fig11.eps,width=27.7pc,height=38pc,angle=0}
%\epsfbox{H1prelim-01-133.fig6.eps}
%\begin{picture}(1,1)
%\put(-320,0){\Large a)}
%\put(-320,-210){\Large b)}
%\end{picture}
\caption{Triple differential dijet cross-section
$d^3\sigma_{ep}/dQ^2 dx_\gamma dy$.}
\label{H2}
\end{figure}
\end{center}
%\end{flushleft}
%================================================================
\section{Conclusions}
The dijet cross-sections measured as a function of \qq, \Et, \xg\ and
\qq, \xg, $y$ at H1 and ZEUS have been presented.

  In the DGLAP evolution scheme, the importance of the \gammaT resolved
photon interactions is clearly demonstrated in the region where
$\EtEQ > Q^2$. Additional \gammaL resolved photon contributions
further improve the agreement of the HERWIG~5.9 prediction with the
measurement.

  Exploring the CCFM approach, the MC program CASCADE~1.0 gives
a qualitative description of the measured differential cross-sections;
however, the \qq dependence is not reproduced. 
On the other hand, %even without introducing a virtual photon structure,
the \xg dependence in CASCADE is comparable to the sum of the direct and
resolved contributions in DGLAP-like MC programs.

%================================================================
%\section*{Acknowledgments}
%Thanks to grant ...


\begin{thebibliography}{99}

%\bibitem{Tana}
%H1 Collab., C. Adloff et al., Eur. Phys. J. {\bf C13} (2000), 397.

\bibitem{chyla}
J.~Ch\'yla and M.~Ta\v{s}evsk\'y,
%``Resolved gamma*(L) in hard collisions of virtual photons: QCD  effects,''
Eur. Phys. J. {\bf C18} (2001), 723.
%, hep-ph/0010254.

%\cite{Chyla:2000hp}
\bibitem{Chyla:2000hp}
J.~Ch\'yla,
%``QCD improved parton distribution functions of gamma*(L),''
Phys.\ Lett.\   {\bf B488} (2000), 289.
%, hep-ph/0006232.

\bibitem{Cascade}
H.~Jung and G.P.~Salam, Eur. Phys. J. {\bf C19} (2001), 351-360.%; \\

\bibitem{Cascade2}
H.~Jung, ``Heavy Quark production at HERA in $k_t$ factorization
          supplemented with CCFM evolution'', These proceedings.

\bibitem{Zeus}
ZEUS Collab., EPS 2001 conference: ``The \qq and \Et dependence 
   of dijet cross sections in $\gamma^*p$ interactions at HERA'',
   paper no. 636.

\end{thebibliography}

\end{document}

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hep-ex0111019.extracted_bib
\begin{thebibliography}{99}

%\bibitem{Tana}
%H1 Collab., C. Adloff et al., Eur. Phys. J. {\bf C13} (2000), 397.

\bibitem{chyla}
J.~Ch\'yla and M.~Ta\v{s}evsk\'y,
%``Resolved gamma*(L) in hard collisions of virtual photons: QCD  effects,''
Eur. Phys. J. {\bf C18} (2001), 723.
%, hep-ph/0010254.

%\cite{Chyla:2000hp}
\bibitem{Chyla:2000hp}
J.~Ch\'yla,
%``QCD improved parton distribution functions of gamma*(L),''
Phys.\ Lett.\   {\bf B488} (2000), 289.
%, hep-ph/0006232.

\bibitem{Cascade}
H.~Jung and G.P.~Salam, Eur. Phys. J. {\bf C19} (2001), 351-360.%; \\

\bibitem{Cascade2}
H.~Jung, ``Heavy Quark production at HERA in $k_t$ factorization
          supplemented with CCFM evolution'', These proceedings.

\bibitem{Zeus}
ZEUS Collab., EPS 2001 conference: ``The \qq and \Et dependence 
   of dijet cross sections in $\gamma^*p$ interactions at HERA'',
   paper no. 636.

\end{thebibliography}