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\begin{document}
\title{Detecting multiparton interactions in minimum-bias events at ALICE} 
\author{Raffaele Grosso$^1$\protect\footnote{\ \ speaker (present affiliation:
Universit\`a di Padova, Italy)},
 Jan Fiete Grosse-Oetringhaus$^2$ for the ALICE collaboration}
\institute{$^1$INFN, Sezione di Torino, Italy\\ 
$^2$CERN -- European Organization for Nuclear Research}
\maketitle
\begin{abstract}
The observed long tail of high-multiplicity events has
questioned the current modelizations for the charged-particle multiplicity
distribution. It has been interpreted as an indirect observation of
multiparton interactions becoming increasingly important at higher collision
energies.  The ALICE detector will measure the
frequency of very high-multiplicity events. The performance for measuring the
charged-particle multiplicity distribution in ALICE is presented.
\end{abstract}

\section{Introduction}
\label{sec:introduction}

Being at LHC the heavy-ion dedicated experiment, ALICE -- A Large Ion
Collider Experiment ~\cite{aliceATlhc} -- has some unique capabilities,
complementary to those of the dedicated p-p experiments. Its 18 detector
systems have been designed to provide high-momentum
resolution as well as excellent Particle Identification (PID) over a broad
momentum range (in particular with very low $p_T$-cutoff) and up to the
highest multiplicities predicted for LHC.

Besides running with Pb ions, the physics programme includes collisions with
lighter ions, lower energy running and dedicated proton-nucleus runs. ALICE
will also take data with proton beams at varying energies, up to the top LHC
energy, to collect
reference data for the heavy-ion programme and to address several QCD topics
for which ALICE is complementary to the other LHC detectors.

The charged-particle multiplicity distribution is among the measurements which
are expected to shed light on the dynamics of multiparton interactions.
We recall here the results of a study for evaluating the performances of
measuring the charged-particle multiplicity distribution with the ALICE detector.

The frequency of non-jet events with very high multiplicity observed by CDF
\cite{cdf2005} has questioned the models for multiparticle production.
Multiparton scattering increases the number of soft particles both in
minimum-bias events and in the underlying event associated with high-$p_t$ jets.
It is expected that multiparton interactions are responsible for the high-multiplicity
tails that break Koba-Nielsen-Olesen (KNO) \cite{kno} scaling and become
significantly more important at LHC energies.
The ALICE detector can make use
of its very low-$p_T$ cutoff ($p_T\approx100$ MeV) and of its
high-multiplicity trigger to investigate the production of large numbers of
soft particles in minimum-bias events.

\section{Multiplicity distribution and multiparton interactions at ALICE}

For p-p and p-\=p collisions at low center-of-mass energies, KNO
scaling describes well the multiplicity distribution. As was first
observed by UA5 (SPS) and E735 (Tevatron) experiments, thus for energies
$\sqrt{s}>200$ GeV, increasing the energy of the collision system leads to
increasingly significant deviations from KNO scaling. This is shown in Figure
\ref{fig:ua5e735}, where it is assumed that the part of the distribution obeying KNO
scaling is due to single-parton interactions, while the deviations are due to
multiparton contributions. In this plot the number of particles $n$ on the x-axis has been
scaled by the average number of particles $\langle n_1 \rangle$, calculated
from the solid curve, obtained by fitting the multiplicity at low energy 
using a polynomial fit in the quantity $x=n/\langle n_1 \rangle$.

\begin{figure}[tb]
\begin{minipage}[b]{0.46\linewidth} 
\centering
\includegraphics[width=6cm]{walkerFig.pdf}
\caption{Comparison of multiplicity distributions at different colliding
energies normalized at their maximum value, taken from \protect\cite{walker}. The
explanation is in the text.}
\label{fig:ua5e735}
\end{minipage}
\hspace{0.4cm} % To get a little bit of space between the figures
\begin{minipage}[b]{0.5\linewidth}
\centering
\includegraphics[width=8cm]{mdFig2.pdf}
\caption{Predictions for the normalized multiplicity distribution in full
phase space for p-p collisions at $\sqrt{s}$ = 14 TeV. The distribution given by the PYTHIA event
generator (red and blue for non-single diffractive and inelastic events
respectively) is compared to a calculation based on the QGSM framework.}
\label{fig:pythiaQGSM}
\end{minipage}
\end{figure}

Among the different explanations of this fact, it has been proposed in the
framework of the Dual Parton Model (DPM) \cite{dpm1,*dpm2} and the Quark Gluon String Model
(QGSM) \cite{qgsm1,*qgsm2} of soft hadronic interactions, that the
parts of the distributions that do not scale are due to multiparton
interactions \cite{matinyan}.

\subsection{Multiplicity analysis}

The ALICE detector will perform measurements of the multiplicity
distribution in pseudorapidity intervals up to $|\eta|<1.4$. We expect that
comparison of model predictions with these measurements will
provide valuable information for understanding multiple particle production
and for tuning the multiparton models included in
different event generators. Figure \ref{fig:pythiaQGSM} compares
the normalized multiplicity distribution for a PYTHIA \cite{pythia1,*pythia2} simulation
\footnote{The version used is 6.2.14 with the so-called "ATLAS tune"
\cite{moraes}.}
to a QGSM model prediction showing
the large inconsistency between the two predictions.

The initial estimate of the multiplicity distribution at ALICE will be
determined by both, counting the SPD tracklets (combination of two clusters in
the two innermost pixel layers) in the region $|\eta|<1.4$,
%\footnote{An SPD tracklet is a
%combination of two clusters, one in each pixel layer, being in line with the
%primary vertex} and by counting the ESD tracks, 
and counting the tracks reconstructed in
the ALICE central barrel, in the region $|\eta|<0.9$. In both cases a set of cuts is
applied for rejecting secondaries.

From full detector simulation one can determine the probability $R_{mt}$ that a
collision with a true multiplicity $t$ is measured as an event with the
multiplicity $m$ and, by varying $t$, one can fill the response
matrix $R$, pictorially shown in Figure~\ref{fig:response}.
In the ideal case of perfect knowledge of the response matrix $R$, and
assuming it to be non-singular, the true multiplicity spectrum $T$ can be
obtained from the measured spectrum $M$ by:
\begin{equation}
 T = R^{-1}M. 
\label{eq:response}
\end{equation}
In practice, the assumptions above do not hold and Eq.~\ref{eq:response} 
generates severe artificial oscillations in the true spectrum; thus unfolding procedures
need to be applied.
Two unfolding procedures have been studied and evaluated for measurements of the
multiplicity distribution in ALICE \cite{JFGOthesis}.
Bayesian unfolding \cite{ago95,*ago99} is an iterative procedure based on the
following equation:
\begin{equation}
 \tilde{R}_{tm} = \frac{R_{mt}P_t}{\sum_{t'}{R_{mt'}P_{t'}}}.
 \label{eq:rtm}
\end{equation}
It relates the conditional probability $\tilde{R}_{tm}$ of a true multiplicity $t$
given a measured value $m$ to the elements of
the response matrix $R_{mt}$ and to the a priori probability $P_t$ for the
true value $t$; at each
iteration the a priori probability is obtained from the following equation:
\begin{equation}
  U_t    = \frac{1}{\epsilon(t)}\sum_m{M_m\tilde{R}_{tm}}.
\label{eq:unfolded}
\end{equation}
As initial a priori distribution the measured one can be used.

The second method, $\chi^2$ minimization, e.g. used in \cite{busku},
consists of finding the unfolded spectrum that
minimizes a $\chi^2$ function measuring the distance between measured and
guessed spectra. It can be expressed by:
\begin{equation}
\chi^2(U)=\sum_m{\left( \frac{M_m-\sum_t{R_{mt}U_t}}{e_m} \right)^2} + \beta P(U)
\end{equation}
where $e$ is the error on the measured spectrum $M$ and $\beta P(U)$
is a regularization term to prevent high-frequency fluctuations.

\subsection{Performance of the unfolding methods}

The performance of the unfolding methods has been evaluated over a rich set of input
distributions to check the behavior of
unfolding for different shapes of the input spectra.

The performance is assessed by calculating the deviation between
input and unfolded distributions in different regions of the distribution.
The free parameters (e.g. the
number of iterations and the weight of the smoothing in the case of the
Bayesian method) have been choosen such that the result is not sensitive to
them.
Furthermore the residuals are evaluated, i.e. the difference
between the measured distribution and the unfolded distribution convoluted
with the response matrix.
Calculating the residuals is an important cross-check which can be performed
also on real data.

\begin{figure}[htb]
\begin{minipage}[t]{0.46\linewidth} 
\centering
\includegraphics[width=6cm]{spd_responsematrix.pdf}
\caption{Detector response matrix visualized by the number of tracklets
found in the SPD vs. the number of generated primary particles in $|\eta|<1$.}
\label{fig:response}
\end{minipage}
\hspace{0.4cm} % To get a little bit of space between the figures
\begin{minipage}[t]{0.5\linewidth}
\centering
\includegraphics[width=8cm]{SystematicsSummary.pdf}
\caption{Summary of the various systematic uncertainties as a function of
multiplicity.}
\label{fig:uncertainties}
\end{minipage}
\end{figure}
   
The comparison of unfolding results obtained with Bayesian unfolding and $\chi^2$
minimization methods has shown that they agree within statistical errors; a
similar comparison should also be performed for real data as a crosscheck
that the unfolding works successfully on the measured data.

\subsection{Systematic uncertainties}

Unfolding using the response matrix is not sensitive to the shape of the
multiplicity distribution, while it might be sensitive to the internal
characteristics of the events and thus to assumptions made in the MC
generator. Also effects like misalignment have an impact on the reconstruction
and thus on the response matrix. Furthermore, the unfolding method itself
causes a non-negligible systematic uncertainty.
An estimate of these uncertainties is summarized in
Figure \ref{fig:uncertainties}, where they are shown as a function of the
multiplicity; the values reported here refer to worst-case scenarios and are
thus expected to reduce improving the knowledge of
the detector (in particular through alignment and calibration) and of the characteristics of the event (like
$p_t$ spectrum and particle abundances). These uncertainties
refer to a specific MC sample and distribution; they will need to be
re-evaluated for the real spectrum.

\section{Summary and conclusions}

The ALICE detector will be able to measure the multiplicity distribution
with high sensitivity in the central barrel rapidity range. Precise
measurements for the different collision systems and colliding energies
included in the ALICE physics programme are expected to contribute clarifying the role of multiparton
interactions in shaping the multiplicity distribution. We expect also that
the multiplicity distribution provided by ALICE will provide a reference
against which models for multiple particle production and their parameters can be validated.
We have presented a procedure for the measurement of the charged-particle
multiplicity distribution with the ALICE detector and the evaluation of its
performance.

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