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\begin{document} \title{Multiple Parton Interactions  \\ at HERA } 
\author{Ll.Mart\'{i} Magro for the H1 and ZEUS Collaborations.
 \protect}
\institute{Universidad Aut\'{o}noma de Madrid, Madrid, Spain.\\ 
}
\maketitle
\begin{abstract}

In lepton-hadron collisions an almost real photon\footnote{\ \ For the virtuality range
considered here.} interacts as a point-like particle as well
as a composite hadron-like system. Event samples with enriched direct- or resolved-photon events are
selected by measuring the photon energy fraction entering in the hard scattering, $x^{obs}_{\gamma}$. This
allows the study of the Underlying Event (UE) and Multiple Parton Interactions (MPI) with a new strategy not
possible at hadron colliders. The H1 collaboration studied photoproduction events with at least two jets
with $P_T^{jets} > 5$~GeV. The highest transverse momentum jet (leading jet) defines four regions in
azimuth: the toward region, defined by the leading jet, the away region,
in the opposite hemisphere and two transverse regions between them, where a measurement of the charged
particle multiplicity is performed and compared to models.


%Two measurements in
%photoproduction are presented. The first looks at three- and four-jet events and the second,
%at the charged particle multiplicity in dijet events in four regions in azimuth defined by the
%jet with highest transverse momentum. 


\end{abstract}

\newcommand{\alps}{\alpha_s}
\newcommand{\alp}{\alpha_s}


%\section{Multiple Parton Interactions}
\section{Introduction}
\label{sec:xxx}

The Underlying Event (UE) can be defined as everything in addition to the lowest order process.
% In quasi-real photoproduction, where $Q^2 \sim 0$, dijet events were selected.

In $ep$ collisions at HERA the mediator boson is a virtual photon. If the virtuality is high the photon
interacts as a point-like particle (direct). At low virtualities the photon may
fluctuate into a quark-antiquark pair and even develop a hadronic structure. In this case, a
parton from the photon interacts with a parton from the proton and only a fraction of the energy
from the photon (resolved) enters in the hard scattering\footnote{\ \ The distinction between direct
and resolved is only unambiguously defined at leading order.}. At HERA, these events can be
selected by measuring the photon energy fraction entering in the hard scattering, $x^{obs}_{\gamma}$. 

Monte Carlo programs (MC) simulate $ep$ collisions with a 2-to-2 parton scattering in leading order
$\alpha_s$. For direct photoproduction, $x^{obs}_{\gamma} > 0.7$, boson-gluon 
fusion is the most important contribution to dijet production. In the event generation,
initial and final state parton radiation and the contributions from the proton remnant are
simulated. Hadronisation models are applied to produce colourless particles. In this
picture, the primary two hard partons lead to two jets while the other parton emissions constitute
the underlying event.

Remnant-remnant interactions are only present when both interacting particles have a composite
structure. This can happen for resolved photon events, $x^{obs}_{\gamma} < 0.7$, via
multi-parton interactions (MPI). By definition, these MPI are part of the UE. Therefore, selecting events
with direct (resolved) photons allows to exclude (include) MPI from the UE. This is an advantage of a
lepton-hadron collider compared to a hadron-hadron collider.

At HERA, three- and four-jet events have been studied~\cite{Tim2} for different $n$-jet invariant mass
regions. Comparisons with ${\cal O} (\alpha \alp)$ matrix element MC programs supplemented with parton
showers and with a ${\cal O} (\alpha \alp^2)$ calculation show that the corrections due to MPI are
needed in order to describe the data. The corrections from MPI are higher for low values of the
invariant mass of the jets. 

%events without any remnant-remnant interaction from those similar to hadron collisions.

The description of MPI in particular and in general of the UE is very important for the LHC physics: Higgs
searches and multi-jet analyses like for the top quark require a proper description of the underlying QCD
aspects. Different MPI models and parton dynamics approaches, however, give very different predictions at
higher energies~\cite{Alekhin:2005dx}. The strategy presented here consists of separating the point-like
from the resolved contributions,~i.e. events with only one remnant from those with two remnants where MPI
are possible. The $ep$ collisions at HERA offer a cleaner environment to study MPI. They can be better
separated from the rest of the UE (parton dynamics, hadronisation, etc) compared to hadron colliders. 


%\subsection{Three- and four-jet events in photoproduction}

%The leading order for an $n$-jet event, via a single chain
%exchange, is ${\cal O} (\alpha \alps^{n-1})$. However, $n$-jet events can also be
%generated via MPI, where several chains are present. Moreover, even soft MPI may affect the
%distribution of multi-jet events by adding or redistributing the energy flow generated by a
%primary process.

\newcommand{\HW}{H\footnotesize{ERWIG}\normalsize{~}}
\newcommand{\PYTH}{P\footnotesize{YTHIA}\normalsize{~}}
\newcommand{\JM}{J\footnotesize{IMMY}\normalsize{~}}
\newcommand{\RAP}{R\footnotesize{APGAP}\normalsize{~}}
\newcommand{\ARI}{A\footnotesize{RIADNE}\normalsize{~}}
\newcommand{\CASC}{C\footnotesize{ASCADE}\normalsize{~}}

%The ZEUS collaboration studied the multi-jet production in the photoproduction regime~\cite{Tim},
%where $Q^2 < 1$ GeV$^2$ and $0.2 < y < 0.85$. Three- and four-jet events, where the jets were
%defined with the $k_T$ clustering algorithm~\cite{k_t-algorithm} and require to have transverse
%energies $E_T^{jets} > 6$ GeV, were studied in the pseudorapidity range $| \eta^{jets}| < 2.4$.
%Furthermore, these events were studied in two different $n$-jet invariant
%mass regions, namely, $25 < M_{n-jets} < 50$ GeV and $M_{n-jets} > 50$ GeV, referred to as the
%low- and high-mass regions, respectively. The cross sections of the
%three- and four-jet low- and high-mass samples were measured differentially.

%The data were compared to predictions from two ${\cal O} (\alpha \alp)$ matrix element MC programs
%supplemented with parton showers, \HW 6.505~\cite{Corcella:2000bw, Corcella:2002jc, Marchesini:1991ch} and
%\PYTH 6.206~\cite{Pythia6.2}, both with and without MPI. In the case
%of \HW, MPI events were simulated via an interface to \JM 4.0~\cite{Butterworth:1996zw}, which is an
%impact parameter dependent model. For \PYTH, MPI were generated using the so-called "simple model"
%~\cite{Pythia6.4}.

%In addition, the three-jet sample was compared to the fixed order
%(${\cal O} (\alpha \alp^2)$) calculation by Klasen, Kleinwort und Kramer~\cite{Klasen:1998cw}. This
%calculation corresponds to the three-jet LO and, at the time, was the highest order prediction
%available in photoproduction. Thus, no calculation was available for the four-jet sample.

%The three- and four-jet cross sections are shown as a function of $M_{n-jets}$ in figure
%~\ref{three_and_four_cross_sections}. In general, both cross sections decrease exponentially with
%increasing $M_{n-jets}$. The \HW and \PYTH predictions with and without MPI are also shown. They are
%normalized to the high invariant $n$-jets mass region ($M_{n-jets} > 50$ GeV), i.e. they are scaled
%to describe the high $M_{n-jets}$ cross section. Both \HW and \PYTH without MPI fail to describe the
%cross section dependence.  When MPI are included, however, they are in quite good agreement with data. The
%discrepancy is larger in the four-jet case. The \PYTH model was run using its default setting
%whereas the \JM model was tuned to the presented data~\cite{Tim}.


%\begin{figure}[ht]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
% begin picture 3 and 4 jets cross sections as fnc of M_{n-jets}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%\centering
%\subfigure{
%\includegraphics[scale=0.35]{Mnj_3j_0_HRW6505_96-00_zufo.eps}
%%\label{delta-phiLX}
%}
%\subfigure{
%\includegraphics[scale=0.35]{Mnj_4j_0_HRW6505_96-00_zufo.eps}
%%\label{delta-phiHX}
%}
%\vspace*{-.3cm}
%\caption[Optional caption for list of figures]{Measured cross section as a function of (a) 
%$M_{3-jets}$ and (b) $M_{4-jets}$ (solid circles). The inner and outer error bars and the
%shaded band represent the statistical, the statistical and the systematic added in quadrature
%and the calorimeter energy scale uncertainties, respectively.}
%\label{three_and_four_cross_sections}
%\end{figure}

%Shown in figure~\ref{three_jets_cross_section}a) is the measured three-jet cross section as a function
%of $M_{3-jets}$, compared to the fixed LO calculation for this process,
%${\cal O} (\alpha \alp^2)$. This calculation was corrected for hadronisation effects and MPI.
%The hadronisation and MPI corrections and their estimated uncertainties are shown
%in~\ref{three_jets_cross_section}b). The hadronisation corrections are constant in $M_{3-jets}$,
%while MPI corrections increase towards low $M_{3-jets}$. The theoretical uncertainties on both
%the MPI corrections and the pQCD predictions are large. The magnitude and the shape of the
%calculation is consistent with the data within the large theoretical uncertainties. This is
%best seen in figure~\ref{three_jets_cross_section}c) where the ratio data over theory is shown.
%Without the large MPI corrections the theoretical description would be far much worse at low
%$M_{3-jets}$.

%\begin{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%            3-jets cross section
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%\begin{center}
%\epsfig{figure=Mnj_3j_0_HRW6505_96-00_zufo_tree.eps,width=5.5cm,height=8.5cm}
%\vspace*{-.7cm}
%\end{center} 
%\caption{{(a) Measured three-jets cross section as a function of $M_{3-jets}$ compared with
%an ${\cal O} (\alpha \alp^2)$ prediction, corrected for hadronisation and MPI effects. (b) The
%hadronisation and MPI correction factors as a function of $M_{3-jets}$. (c) The ratio of the
%$M_{3-jets}$ cross section divided by the theoretical prediction. The theoretical uncertainty
%is represented by shaded bands.
%\label{three_jets_cross_section} }}
%\end{figure}



\section{Charged particle multiplicity in photoproduction}

MPI and its contribution to the UE were studied by the H1
collaboration~\cite{LluisPrelim, MartiMagro:2009zz} using dijet photoproduction.
Events with $Q^2 < 0.01$~GeV$^2$ and 0.3 $< y <$~0.65 were selected. The jets were defined applying the
inclusive $k_t$-jet cluster algorithm~\cite{ktAlgo} in the laboratory frame. The jets were required to
have transverse momentum $P_T^{jets} > 5$ GeV and pseudo-rapidity $|\eta^{jets}| <$~1.5. Within these
events, charged particles with transverse momenta $P_T^{track} > 150$~MeV in the range
$|\eta^{track}| <$~1.5 were selected. 

The analysis procedure, inspired by the CDF collaboration~\cite{Acosta:2001rm}, is the following:

Four regions in the azimuthal angle, $\phi$, were defined with respect to the leading jet as indicated in
figure~\ref{regions}. The leading jet defines the azimuthal angle, $\phi =$ 0. The region
$|\phi| <$ 60$^\circ$ is defined as the toward region and is expected to contain all particles from the
leading jet. The away region is defined by $|\phi| >$ 120$^\circ$ which often contains the second leading
jet and most of its particles to balance the transverse momentum in the event. In the transverse regions,
60$^\circ < |\phi| <$ 120$^\circ$, the contribution from the primary collision is usually small and thus
the effects from the UE should be most visible.

\begin{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%            4-regions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{center}
\epsfig{figure=azimuth_regions.eps,width=6.5cm,height=6.3cm}
\vspace*{-.7cm}
\end{center} 
\caption{{Definition of the four azimuthal regions. The toward region is defined by the leading jet
and by this means defines the away and transverse region. The scalar sum of the transverse momenta
$P_T^{sum} = \sum_i^{tracks} P_T^i$ is calculated event by event in each transverse region. This
defines the high and low activity transverse region.
\label{regions} }}
\end{figure}

In the transverse regions, a high activity and a low activity region are defined event by event depending
on which region contains the higher scalar sum of the transverse momentum of charged particles, 
$P_T^{sum} = \sum_i^{tracks} P_T^i$. The high activity region is more affected by higher order QCD
contributions than the low activity region by definition: if higher order radiation is emitted this will
increase the $P_T^{sum}$ in that transverse hemisphere.

%if in the transverse region hard radiation is emitted due to higher order contributions this will increase 
%the $P_T^{sum}$ in that transverse region hemisphere.

The average charged particle multiplicity, $\langle N_{charged} \rangle$, as a function of the transverse
momentum of the leading jet, $P_T^{Jet_1}$, for the different azimuthal regions is shown in
figures~\ref{towawa_py}-\ref{trans_casc}. The measurement is performed for resolved and a direct
photon enriched events, i.e. $x^{obs}_{\gamma} < $ 0.7 and $x^{obs}_{\gamma} > $ 0.7, respectively.

The $\langle N_{charged} \rangle$ distributions are corrected to the level of charged stable hadrons
using an iterative Bayes unfolding method (see \cite{unfolding}). They are
compared to two MC predictions: \PYTH~\cite{Sjostrand:2006za}
and \CASC~\cite{Jung:2000hk, Jung:2001hx}, both implement leading order in $\alp$ matrix elements. The matrix
elements are supplemented with initial and final state radiation according to the DGLAP evolution equations
in \PYTH and the ones of CCFM in \CASC. In \PYTH a model of MPI is available for $ep$ collisions. \CASC uses unintegrated gluon density functions (updf) and off-shell matrix elements. It does not include the resolved component of the photon and has not model for MPI implemented.  In \PYTH the CTEQ 6L~\cite{Pumplin:2002vw}
pdf was used while in \CASC set2 and set3~\cite{Hansson:2003xz} were used.

In the toward and away regions $\langle N_{charged} \rangle$ increases with the $P_T^{Jet_1}$ by about
30$\%$ from the lowest to the highest $P_T^{Jet_1}$ bin. On the contrary, in the transverse regions the
multiplicity tends to decrease although the effect is much weaker. In the toward regions the particle
multiplicity is slightly higher than in the away regions but in the transverse high activity regions
the multiplicity is much higher than in the low activity regions. The multiplicity is higher for resolved
enriched than for direct enriched events.

\begin{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%            Toward & Away: PYTHIA
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{center}
\epsfig{figure=charged_towardaway_py_090320.eps,width=12.5cm,height=9.cm}
\vspace*{-.7cm}
\end{center} 
\caption{{Average charged particle multiplicity as a function of the transverse momentum of
the leading jet, $P_T^{Jet_1}$, in the toward and away regions and for the low and high $x^{obs}_{\gamma}$
sub-samples.
\label{towawa_py} }}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%            Toward & Away: CASCADE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{center}
\epsfig{figure=charged_towardaway_casc_090320.eps,width=12.5cm,height=9.cm}
\vspace*{-.7cm}
\end{center} 
\caption{{Average charged particle multiplicity as a function of the transverse momentum of
the leading jet, $P_T^{Jet_1}$, in the toward and away regions and for the low and high $x^{obs}_{\gamma}$
sub-samples.
\label{towawa_casc} }}
\end{figure}

\begin{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%            Transverse: PYTHIA
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{center}
\epsfig{figure=charged_trans_py_090320.eps,width=12.5cm,height=9.cm}
\vspace*{-.7cm}
\end{center} 
\caption{{Average charged particle multiplicity multiplicity as a function of the transverse momentum of
the leading jet, $P_T^{Jet_1}$, in the toward and away regions and for the low and high $x^{obs}_{\gamma}$
sub-samples.
\label{trans_py} }}
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%            Transverse: CASCADE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\begin{center}
\epsfig{figure=charged_trans_casc_090320.eps,width=12.5cm,height=9.cm}
\vspace*{-.7cm}
\end{center} 
\caption{{Average charged particle multiplicity multiplicity as a function of the transverse momentum of
the leading jet, $P_T^{Jet_1}$, in the toward and away regions and for the low and high $x^{obs}_{\gamma}$
sub-samples.
\label{trans_casc} }}
\end{figure}

In figures~\ref{towawa_py} and~\ref{towawa_casc} the data are compared to different MC predictions in the
toward and away regions. The \PYTH MC describes data quite well if contributions from MPI are included in
the simulation (figure~\ref{towawa_py}). The contributions from MPI decrease as $P_T^{Jet_1}$ grows according
to this model. The \CASC MC describes the data fairly well. For direct enhanced events,
$x^{obs}_{\gamma} > $ 0.7, \CASC describes the data perfectly. For resolved enhanced events,
$x^{obs}_{\gamma} < $ 0.7, however, the predicted multiplicity is lower than in data, especially
at low $P_T^{Jet_1}$.


Figures~\ref{trans_py} and~\ref{trans_casc} show a comparison between data and the MC predictions in the
transverse regions. Like in the toward and away regions, including MPI improves the description of the data
in all bins for \PYTH\footnote{\ \ \PYTH describes the data only when including MPI. For more details see~\cite{LluisPrelim, MartiMagro:2009zz}}. In both $x^{obs}_{\gamma} > $ 0.7 transverse regions (b and d) \PYTH + MPI and
\CASC describe the data well. However, they somewhat underestimate the data in the resolved enriched transverse regions. Here, the
shape predicted by \PYTH + MPI follows the data distribution, although the absolute value of the multiplicity
is slightly too low. \CASC predicts an even lower multiplicity in these regions but it is much better than
\PYTH without MPI, although \CASC does not include a resolved component and any MPI model. The description
of \CASC is better in the high activity region, where higher order corrections are more important, than
in the low activity region, which is expected to be most sensitive to MPI. These discrepancies decrease with increasing $P_T^{Jet_1}$.


%Figures~\ref{trans_py} and~\ref{trans_casc} show a comparison between data and the MC predictions in the
%transverse regions. In both $x^{obs}_{\gamma} > $ 0.7 regions (b and d) \PYTH + MPI and \CASC describe the
%data well. However, in the resolved enriched region they somewhat underestimate the data, particularly \CASC in
%the low activity region. Like in the toward and away regions \PYTH + MPI improves the description of the data
%in all bins. Although \CASC does not include a resolved component in the simulation, its description of the
%transverse $x^{obs}_{\gamma} < $ 0.7 high activity region (a) is much better than the one given by \PYTH without
%MPI. The \CASC description is fairly good compared to the \PYTH + MPI description although it does not include
%any MPI model. The discrepancies with the data decrease with increasing $P_T^{Jet_1}$. In the low activity
%region (c) the shape predicted by \PYTH + MPI follows the data distribution although the absolute value of the
%multiplicity is slightly too low. \CASC predicts an even lower multiplicity in this region.





\section{Conclusion}

%In both analyses, contributions from MPI are suggested. The three- and four-jet photoproduction cross
%sections shapes cannot be described by the
%${\cal O} (\alpha \alp)$ plus parton shower calculations. In the three-jet case the LO pQCD
%calculation needs to include the estimated hadronisation and MPI effects to describe data, where
%the latter has a large contribution towards low $M_{3-jets}$.

The average charged particle multiplicity in dijet photoproduction has been
measured as a function of $P_T^{Jet_1}$ in four regions of the azimuthal angle $\phi$: the toward, away,
transverse high and low activity regions. The data have been investigated for enhanced photon point-like
interactions with the proton events and enhanced photon resolved events. The data have been compared to
predictions of the \PYTH and \CASC MC generators. 
%The charged particle multiplicity has been investigated for two different regions in $x^{obs}_{\gamma}$ 
%separating the point-like from the hadronic (resolved).

\PYTH without MPI does not produce enough particles, especially at low $x^{obs}_{\gamma}$ and in the
transverse regions. Including MPI leads to a good description of the data.

\CASC provides a good description of the data in the high $x^{obs}_{\gamma}$ regions. In the low
$x^{obs}_{\gamma}$ regions it produces too few particles, especially in the low activity region.

%The toward and away regions are fairly well described by \CASC and only at low $x^{obs}_{\gamma}$ and low
%$P_T^{Jet_1}$ \CASC does not produce enough particles. The transverse region at high $x^{obs}_{\gamma}$ is 
%well described by \CASC. The particle multiplicity predicted by \CASC is somewhat too low in the high 
%activity region for resolved enriched photon events although larger than \PYTH without MPI. The
%differences with data decrease with increasing $P_T^{Jet_1}$. 

\CASC describes the data better than \PYTH without MPI both at low $x^{obs}_{\gamma}$ and at high
$x^{obs}_{\gamma}$, where contributions from MPI are smaller. The discrepancies of \CASC
with the data in the high activity region are smaller than in the low activity region, the
former is expected to be more sensitive to higher orders and the later to MPI. This points
to a possible better parton dynamics approach in \CASC which could be important in the determination
of the amount of MPI. Reducing the amount of MPI needed to describe the data, by improving the parton
dynamics in the pQCD regime, would reduce the theoretical uncertainty for the description of
MPI. This would have important benefits for physics predictions at LHC energies.

%\CASC describes the direct photon enrich region. In the resolved region some activity is missing although 
%higher than \PYTH without MPI at low $P_T^{Jet_1}$ especially in the high activity region and more clearly in 
%the low activity region. 

%In the region of $x^{obs}_{\gamma} > 0.7 $ the charged particle multiplicity is reasonably well
%described by \PYTH and \CASC. However, in the case of \PYTH the charged particle production would be
%somewhat too low without including MPI.
 
%The region of $x^{obs}_{\gamma} < 0.7 $ , that the charged particle multiplicity is
%larger than what is expected from this simulation, and is well described when multi-parton
%interactions are included. 
%The multi-parton interaction simulation describes the measurement better when the colour
%structure of the final state is chosen such that long string lengths are preferred.

%It was also shown, that the charged particle multiplicity in the transverse region 
%cannot be accounted for by increasing the amount of initial and final state radiation.

%Here comes the text, with an example citatation \cite{feyn} of a famous book.
%And all of this is typestted with the style of {\tt cernrep} supplemented with the citation option {\tt mcite}, where
%an exapmple is shown here using the command for multiple citations \cite{art1,*art2,*art3}.
%\subsection{How to \LaTeX\ it}

%The references are optionally included via \bibtex . To produce the \LaTeX\
%document, do the following:
%\begin{verbatim}
%latex mpi08.tex
%bibtex mpi08 
%latex mpi08 
%latex mpi08 
%dvips mpi08
%\end{verbatim}
%The \bibtex\ file \verb+mpi08.bib+ can be edited 

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

This work has been partially supported by the Spanish Consolider-Ingenio 2010
Programme CPAN (CSD2007-00042).

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