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\begin{document} \title{Minimum Bias Physics at the LHC with the ATLAS Detector} 
\author{W. H. Bell on behalf of the ATLAS Collaboration}
\institute{Universit\'e~de~Gen\`eve, Section~de~Physique, 24~rue~Ernest~Ansermet, CH-1211~Geneve~4} 
\maketitle

\begin{abstract}
  This paper presents the status of Minimum Bias physics analysis with
  the ATLAS detector~\cite{:2008zzm}.  The current uncertainties in
  modelling soft p-p inelastic collisions at LHC energies are
  discussed in the context of primary charged track measurements. The
  selection and reconstruction of inelastic p-p interactions with the
  ATLAS detector at the LHC is discussed.  The charged track
  reconstruction performance is explored using a
  GEANT4~\cite{Agostinelli:2002hh} simulation of the ATLAS detector.
\end{abstract}

\section{Introduction}

The properties of inelastic proton-proton and proton anti-proton
interactions have previously been studied over a wide range of
energies~\cite{Matinyan:1998ja,Alexopoulos:1998bi,Ansorge:1988fg,Ansorge:1988kn,Alner:1987wb,Ansorge:1986xq,Abe:1989td,Abe:1988yu,Breakstone:1983ns,Arnison:1982ed}.
Previous analyses have selected events with minimal bias and
illustrated their behaviour through $dN_{Ch}/d\eta$, $dN_{Ch}/dp_T$,
KNO scaling~\cite{Koba:1972ng}, and $\langle p_T \rangle$~vs~$N_{Ch}$
distributions.  These distributions are typically produced from a
selection of non-single-diffractive events, defined as a sample of
inelastic events, where the trigger acceptance for single diffractive
events is very low.  Previous results from CERN and Fermilab
experiments have been used to tune~\cite{Moraes:2007rq} the
PYTHIA~\cite{Sjostrand:2006za} event generator such that the generator
properly describes previous measurements.  In particular
figure~\ref{figure:eta0_prediction} illustrates the measured and
predicted charged particle density for non-single diffractive events
as a function of the centre of mass energy.  There are clear
differences in the predicted multiplicities of PYTHIA(ATLAS and CDF
tune-A~\cite{CDFtuneA}) and PHOJET~\cite{phojet-thesis,phojet-web} at
the Large Hadron Collider (LHC) centre of mass energies of $\sqrt{s} =
10$~TeV and $\sqrt{s} = 14$~TeV.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[width=8cm]{figures/rho-tuned.eps}
  \end{center}
  \caption{Central charged particle density for non-single diffractive 
inelastic $p$-$\bar{p}$ collisions. \label{figure:eta0_prediction}}
\end{figure}

The first physics run of the Large Hadron Collider (LHC) is expected
to take place in late 2009.  The LHC is expected to run first at a
centre of mass energy $\sqrt{s} = 10$~TeV during 2009-2010, and then at
$\sqrt{s} = 14$~TeV following a shutdown.  Data collected from the first
physics run at the LHC will allow models of soft QCD processes to be
constrained.  These studies are vital to understand QCD within the LHC
energy regime and to model additional proton-proton interactions,
which will be abundant at higher instantaneous luminosities.


\section{Predicted Properties}

The total p-p cross section can be expressed as a sum of the
components parts,
\[\sigma_{tot} = \sigma_{elas} + \sigma_{sd} + \sigma_{dd} + \sigma_{nd}\]
where these cross-sections are elastic ($\sigma_{elas}$), single
diffractive ($\sigma_{sd}$), double diffractive ($\sigma_{dd}$) and
non-diffractive ($\sigma_{nd}$), respectively.  In this approximation
the small central diffractive component of the cross
section is ignored.  Predictions for the cross sections at 14~TeV are given
elsewhere~\cite{Aad:2009wy}.  10~TeV cross sections are expected to be
of the order of 10\% lower.  Using the PYTHIA and PHOJET event
generators the predicted properties of the $dN_{Ch}/d\eta$ and
$dN_{Ch}/dp_T$ distributions are illustrated in
figure~\ref{figure:evgen}.  This paper focuses on the $dN_{Ch}/d\eta$
and $dN_{Ch}/dp_T$ distributions, and the very first results expected
from the ATLAS detector.

\begin{figure}[h!]
\begin{center}
\begin{tabular}{cc}
\resizebox{0.5\textwidth}{!}{\includegraphics{figures/dndeta_all_pythia_phojet.eps}}
&
\resizebox{0.5\textwidth}{!}{\includegraphics{figures/dndpt_all_pythia_phojet.eps}}
\\
(a) & (b)
\end{tabular}
\caption{Pseudorapidity (a) and transverse momentum (b) distributions  of stable charged
  particles from simulated 14~TeV $p$-$p$ inelastic collisions generated using 
  the PYTHIA and PHOJET event generators}
\label{figure:evgen}
\end{center}
\end{figure}

\section{Event Selection}

The LHC is expected to run with a range of different operating
parameters, providing different mean numbers of interactions per p-p
bunch crossing.  During initial running it is expected that the mean
number of inelastic interactions per p-p bunch crossing will be much
less than one.  Within this operating regime it is necessary to select
the rare events containing inelastic interactions over those where the
beams do not produce such an interaction and only detector noise is
recorded.  Once the mean number of interactions approaches, or exceeds
unity the majority of inelastic interactions will be selected by
simply requiring the presence of two crossing proton bunches.

The ATLAS detector~\cite{:2008zzm} is a multi-purpose detector
designed to study all areas of physics at the LHC.  The key components
for early Minimum Bias physics measurements are the Inner Detector
(ID) and sections of the trigger system dedicated to the selection of
inelastic interactions with minimal bias.  The ID covers radii of
50.5~mm to 1066~mm and is composed of a silicon pixel system, a silicon
micro-strip tracker (SCT), and a gas-based transition radiation
detector (TRT).  The ID is housed inside a solenoid magnet which
produces a 2~Tesla axial magnetic field.  A summary of the
active ID acceptance is given in table~\ref{table:id}, where the
silicon tracking detectors and the TRT cover $|\eta| < 2.5$ and
$|\eta| < 2.0$ respectively.

\begin{table}[h!]
  \begin{center}
    \begin{tabular}{|l||c|c|c|} \hline
       & & Radius (mm) & Length (mm) \\ \hline
      \textbf{Pixel} & Barrel (3 layers) & $50.5 < R < 122.5$ & $0 < |z| < 400.5$ \\
        & End-cap (2x3 disks) & $88.8 < R < 149.6$ & $495 < |z| < 650.0$ \\ \hline
      \textbf{SCT} & Barrel (4 layers) & $299 < R < 514$ & $0 < |z| < 749$ \\
        & End-cap (2x9 layers) & $275 < R < 560$ & $839 < |z| < 2735$ \\ \hline
      \textbf{TRT} & Barrel (73 straw planes) & $563 < R < 1066$ & $0 < |z| < 712$ \\
        & End-cap (160 straw planes) & $644 < R < 1004$ & $848 < |z| < 2710$ \\ \hline
    \end{tabular}
  \end{center}
  \caption{A summary table of the ATLAS ID acceptance.}
  \label{table:id}
\end{table}

During initial low luminosity running, events will be selected with the
Minimum Bias trigger.  For initial measurements based on charged
tracks reconstructed in the ID, inelastic collisions will be selected
with either the Minimum Bias ID trigger or the Minimum Bias Trigger Scintillators
(MBTS).  The primary Minimum Bias ID trigger uses Pixel clusters and SCT space
point information and covers $|\eta| < 2.5$.  The MBTS are situated at
$z = \pm 3560$~mm and are segmented into eight units in azimuth and two
units ($2.82 < |\eta| < 3.84$, $2.09 < |\eta| < 2.82$) in
pseudo-rapidity.

The ATLAS detector has a three stage trigger to select events: Level
1, Level 2 and the Event Filter (EF).  Inelastic events are selected
if they satisfy one of the Minimum Bias Level 1 triggers.  Most of the
events containing tracks within the ID acceptance will be selected by
either the level 1 MBTS trigger or the random filled bunch trigger
(L1\_RDO\_FILLED).  Events passing the random filled bunch trigger
will be filtered at Level 2 by using ID information.

A Level 1 MBTS trigger is formed by requiring a given number of MBTS
counters above threshold.  For the selection of inelastic interactions
with minimal bias it is necessary to require a minimum number of MBTS
counters.  A requirement of just one counter is sensitive to the
electronic noise level, and therefore two counters are preferred.  For
minimum trigger selection bias the number of MBTS counters above
threshold from the two sides are summed.  If this sum is greater than or
equal to 2 the primary physics trigger L1\_MBTS\_2 fires.

L1\_RDO\_FILLED simply requires the presence of two proton bunches and
a random clock cycle.  During the initial running period the
luminosity is expected to be too low for L1\_RDO\_FILLED to be used
without filtering at Level 2.  At Level 2, events selected with \\
L1\_RDO\_FILLED are passed to the Minimum Bias ID trigger, where the
total number of Pixel clusters and SCT space points are used to select
p-p bunch crossings containing an inelastic interaction.  The SCT
modules are made from pairs of silicon sensors mounted in small angle
stereo with each other.  A space point is formed from a strip hit
coincidence of the pair of sensors, reducing the sensitivity to noise.
Pixel clusters are formed from a cluster of pixels above a
time-over-threshold constraint.  While the Pixel clusters only include
one sensor plane the noise occupancy is expected to be low
enough~\cite{Dobos:2007zz,Aad:2008pn} for the total number of pixel clusters to
be used within the Minimum Bias ID trigger.  Thresholds for the
multiplicity constraints on Pixel and SCT detectors were set by
studying the simulated performance of these detectors, where the noise
model contained random electronic noise occupancies taken from
detector measurements.  Events which pass the Pixel cluster and SCT
space point requirements are further filtered at the EF by requiring a
number of reconstructed tracks.  For the EF selection of an event two
tracks were required to have $p_{T} > 200$~MeV and a nominal $|Z_{0}| <
200.0$~mm, minimally biasing the selection, but rejecting some beam
background.

Inelastic single diffractive, double diffractive and non-diffractive
events were generated with the PYTHIA event generator at $\sqrt{s} =
14$~TeV.  These events were then passed through a GEANT4 simulation
and overlaid with simulated detector noise.  In addition to the
physics samples, beam-gas was simulated with the HIJING event
generator~\cite{Gyulassy:1994ew} and events containing no p-p
interactions were also studied.  The resulting Minimum Bias trigger
efficiencies are illustrated in figure~\ref{figure:trigEff}, where the
Pixel and SCT efficiencies were calculated with the other multiplicity
requirement set to zero, the track trigger efficiency includes the
prior selection of Pixel and SCT at Level 2, and the simulated
noise in the MBTS is artificially high within the beam-gas sample.

\begin{figure}[h!]
\begin{center}
\begin{tabular}{cc}
\resizebox{0.5\textwidth}{!}{\includegraphics{figures/triggerEffMBTS2.eps}}
&
\resizebox{0.5\textwidth}{!}{\includegraphics{figures/triggerEffPixelSP.eps}}
\\[-3mm]
(a) & (b) \\
\resizebox{0.5\textwidth}{!}{\includegraphics{figures/triggerEffSctSP.eps}}
&
\resizebox{0.5\textwidth}{!}{\includegraphics{figures/triggerEffSpTrackTrigger.eps}}
\\[-3mm]
(c) & (d) 
\end{tabular}
\caption{The trigger efficiency for: MBTS\_2 as a function of the counter threshold (a), Pixel space point as a function of the number of Pixel space points required (b), SCT space point as a function of the number of SCT space points required (c), and track trigger as a function of the required number of reconstructed tracks (d).}
\label{figure:trigEff}
\end{center}
\end{figure}

Using a nominal MBTS signal threshold of 40~mV from previous cosmic
studies, and SCT and Pixel thresholds defined by the requirement of a
maximum detector noise trigger efficiency of $5 \times 10^{-4}$, the trigger
efficiencies were calculated and are listed in table~\ref{table:trigEff}.

\begin{table}[h!]
\begin{center}
\begin{tabular}{|l||c|c|c|c|} \hline
 & MBTS\_1\_1 & MBTS\_2 & SP & SP \& 2 Tracks \\ \hline
Non-diffractive & 99\% & 100\% & 100\% & 100\% \\
Double-diffractive & 54\% & 83\% & 66\% & 65\% \\
Single-diffractive & 45\% & 69\% & 57\% & 57\% \\
Beam-gas & 40\% & 54\% & 47\% & 40\% \\ \hline
\end{tabular}
\caption{A table of trigger efficiencies for: an MBTS threshold of
  40~mV, requirements of $\geqq 12$ Pixel and $\geqq 3$ SCT space points (SP), and the
  requirement of two tracks with nominal $Z_{0} < 200$~mm and $p_{T} >
  200$~MeV after the SP requirement.}
\label{table:trigEff}
\end{center}
\end{table}

\section{Event Reconstruction}

Initial Minimum Bias physics measurements involve the reconstruction
of charged particle multiplicity distributions.
Figure~\ref{figure:evgen} clearly illustrates the predicted event
properties, where the most probable particle $p_T$ is expected to be
around 220~MeV.  In high multiplicity environments, such as expected in
high energy hard scatter processes or higher luminosity running at the
LHC, it is necessary to normally require a $p_T$ cut-off of 500~MeV or
higher within the track reconstruction software.  This cut-off is
required to reduce the number of combinations of track candidates and
improve the performance of the track reconstruction algorithms.  For
Minimum Bias events a second low $p_T$ track reconstruction step has
been introduced.

The ATLAS track reconstruction software~\cite{Cornelissen:2008zzc} is
run over the silicon hits twice: finding tracks with $p_T$ above
500~MeV and then reconstructing the remaining tracks down to a minimum
$p_T$ of 100~MeV.  Hits that are attached to tracks reconstructed
during the first tracking pass are tagged such that they are not used
during the second pass.  Then the second reconstruction pass runs with
a wider azimuthal road size and looser track reconstruction
constraints.  During both the first and second pass of the track
reconstruction the silicon tracks are projected into the TRT, finding
track extensions where present.  The combined tracking performance was
studied from inelastic non-diffractive p-p events generated using
PYTHIA at $\sqrt{s} = 14$~TeV which were passed through the ATLAS GEANT4
detector simulation.

Following previous tracking performance
studies~\cite{Cornelissen:2008zz}, the track reconstruction efficiency
was defined as the ratio of reconstructed tracks matched to Monte
Carlo particles, divided by all stable charged primary Monte Carlo
particles.  The fake rate was defined as the ratio of all primary
reconstructed tracks not matched to a Monte Carlo particle divided by
all primary reconstructed tracks.  For the measurement of tracking
efficiency and fake rate, primary Monte Carlo particles and primary
reconstructed tracks were selected by requiring: \\
%
{\small
\begin{itemize}
\item ID Acceptance ($|\eta| < 2.5$) 
\item Primary Particle \\ [-0.75ex]
  \begin{itemize} 
  \item Not generated from the GEANT4 simulation.
  \item $|d_{0}| < 2$~mm with respect to the generated primary vertex. \\ [-0.75ex]
  \end{itemize}
\item Stable charged particle PDG id. 
\item $p_{T} > 100$~MeV
\end{itemize}
}
%
\noindent and
%
{\small 
\begin{itemize}
\item ID Acceptance ($|\eta| < 2.5$) 
\item N$^{o.}$ Silicon Hits $\geqq 5$, from 11 planes of silicon. 
\item Primary Track \\ [-0.75ex]
  \begin{itemize}
  \item $|d_{0}| < 2$~mm with respect to the generated primary vertex. 
  \item $|Z_{0} sin(\theta)| < 10$~mm \\ [-0.75ex]
  \end{itemize}
\item $p_{T} > 100$~MeV \\
\end{itemize}
}
%
\noindent respectively.  The resultant tracking performance is illustrated in 
figure~\ref{figure:trackingPerformance}.


\begin{figure}[h!]
\begin{center}
\begin{tabular}{cc}
\resizebox{0.5\textwidth}{!}{\includegraphics{figures/effPtAll_bi_err.eps}}
&
\resizebox{0.5\textwidth}{!}{\includegraphics{figures/effEtaAll_bi_err.eps}}
\\
(a) & (b) \\
\resizebox{0.5\textwidth}{!}{\includegraphics{figures/fakesPtAll_bi_err.eps}}
&
\resizebox{0.5\textwidth}{!}{\includegraphics{figures/fakesEtaAll_bi_err.eps}}
\\
(c) & (d) 
\end{tabular}
\caption{Tracking performance: efficiency as a function of $p_T$ (a) and pseudo-rapidity (b), normalised track fakes as a function of $p_T$ (c) and pseudo-rapidity (d).}
\label{figure:trackingPerformance}
\end{center}
\end{figure}

\section{Conclusion}

The ATLAS Collaboration expects to record the first p-p inelastic
collisions later this year.  A trigger system to select inelastic
events with minimal bias within the tracking acceptance has been
developed.  The trigger performance given in table~\ref{table:trigEff}
indicates good acceptance of inelastic events suitable for minimum
bias physics studies.  Reconstruction of inelastic non-diffractive
events has been explored with low momentum track reconstruction
algorithms.  The performance of these low momentum track
reconstruction algorithms is illustrated in
figure~\ref{figure:trackingPerformance}, and clearly demonstrates
track reconstruction below the nominal 500~MeV cut-off.  Further
improvements in the track reconstruction efficiency and reduction of
the associated fake rates are expected following additional algorithm
tuning.  Previous studies~\cite{Aad:2009wy} have found the systematic
uncertainty on an expected $dN/d\eta$ measurement to be of the order
of 8\% for a non-single-diffractive measurement, sufficient to
distinguish between different theoretical models.  The ATLAS
Collaboration therefore looks forward to the first LHC beam and the
first physics results.

\section*{Acknowledgements}

The work was undertaken within the framework of the ATLAS
Collaboration.  This paper presents results that were obtained using
the ATLAS collaboration GEANT4 simulation, Inner Detector track
reconstruction software, and detector noise measurements.  The author
therefore thanks all of those involved in these areas of research.

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