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\begin{center}
\mbox{\Huge \bf ATLAS NOTE}
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\mydocversion
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% \vspace*{-1cm}
\title{Performance of the ATLAS Muon Trigger Slice with Simulated Data}
\author{ATLAS Collaboration}
%
% The abstract
%
\begin{abstract}

The overall functionality and performance of the muon
trigger system with respect to data produced as part of the ATLAS
Computing System Commissioning effort is described.
%   The muon slice is composed of three trigger levels: an hardware implemented
%Level1 (L1) and a software implemented High Level Trigger (HLT) composed
%of Level2 (L2) and Event Filter (EF).
%   L1 uses the full granularity of the RPC and TGC to selects muons with
%transverse momentum above six programmable thresholds with a coarse evalua-
%tion of the muon direction, and associates the trigger candidate with the correct
%LHC bunch crossing. L2 algorithms run on a subsample of the event, determined by the RoI selected by L1,
%producing a feature data object containing muon measured quantities that will be used to perform the trigger decision.
%In the EF, offline muon reconstruction algorithms, adapted to work in the HLT
%framework, accesses and reconstructs the full event using more complex proce-
%dures and using offline services.
The physics performance in terms of trigger efficiency and accepted
rates is studied for the muon inclusive signatures for different
luminosity scenarios. Dedicated studies on physics samples with
single and double muon final states
are also performed in order to evaluate the trigger efficiencies
on realistic data and background rejection capabilities. Methods to
evaluate muon trigger efficiencies from real data are discussed.
%Furthermore, strategies to use muon signals in calorimeters to select isolated muons and
%for tagging muon in Tile are presented together with results
%for  efficiency and background contamination.
Furthermore, strategies to use the ATLAS calorimeters to tag and
select isolated muons are presented.
\end{abstract}
%
\end{titlepage}
%
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% Introduction
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\tableofcontents

\newpage
\pagestyle{empty}
\section{Introduction}

Triggering and identifying muons will be crucial for many LHC physics analyses.
%The possibility of identifying and triggering muons  is crucial at LHC for most physics studies.
In accordance with the ATLAS general trigger scheme,  the muon
trigger system has three distinct levels: L1, L2, and the Event
Filter (EF).
%agb comment
%Each trigger level refines the
%decisions made at the previous level and, where necessary, applies
%additional selection criteria. L1 uses a limited amount of the total
%detector information to make a decision in less than 2.5$\mu$s,
%reducing the rate to about 75 kHz.
%agb end
%The two higher levels access more detector information for a
%final rate of up to 200 Hz with an event size of approximately 1.5 Mbyte.
%The L1 trigger searches for high transverse-momentum muons
%using trigger chambers in the barrel and end-cap regions of the muon spectrometer.
%L1 is processed by the muon central trigger processor
% which combines the information from the two trigger system.
%The L1 trigger also provides Regions of Interest (RoI), i.e. geographical coordinates in
%$\eta$ and $\phi$, of those regions within the detector
%where its selection process has identified interesting object in the event.
%In addition, the RoIs also contain information about the \pt\ threshold of the
%muon passed.
%This information is subsequently used by the High Level Trigger (HLT).
%agb com
%The L2 selection is seeded by the Regions of Interest (RoI)
%information provided by the L1 Trigger. All detector data within the
%L1 RoI are available for processing at L2.
%agb end
%The L2 selection is seeded by the RoI information provided by the L1 trigger and  uses,
%at full granularity and precision, all available detector data within
%the RoI.
%agb com
%L2 is designed to reduce the trigger rate to approximately 3.5 kHz,
%for an average event processing time of approximately 10 ms.
%The final stage of the event selection is provided by the EF,
%which reduces the event rate to about 200 Hz.
%Its selection is implemented using offline analysis procedures and will have an average
%event processing time of order 1 s.
%agb end
The paper discusses the software tools used for muon trigger
reconstruction and the algorithm selection strategy and trigger
configuration. Next, the resolution and selection efficiencies of
the various muon triggers are presented, followed by a discussion of
the trigger rates for various luminosities. Subsequently, the
rejection of background from in-flight meson decays and selection of
isolated muons using calorimeter information is discussed. Finally,
the trigger performance on the di$-$muon final states \Zmm
%$Z \rightarrow %\mu\mu$
and $Z^\prime \rightarrow \mu\mu$ is presented along with a
description of plans to determine the trigger efficiency from
collider data.
% This paper is structured as follows. In the next Section
%the simulated samples are listed along with a short description of
%the software tools used for the muon trigger reconstruction. Section
%\ref{slice}  describes the
% algorithm selection strategy and the muon trigger configuration.
%In Sections \ref{sec:l1_perf}, \ref{sec:l2_perf}, and
%\ref{sec:ef_perf} the resolution and selection efficiencies of the
%various muon triggers are presented. In Section \ref{sec:rates} the
%rates of the various muon trigger thresholds for a startup
%luminosity of \begL\ and nominal luminosity scenarios of \lowL\ and
%\highL\ are presented. The rejection of background from in-flight
%meson decays is addressed in Section \ref{EF_pi_K}. Selection of
%isolated muons with calorimeters is discussed in Section .
%Section \ref{sec:muiso_perf} presents a discussion of analysis for selecting
%isolated muons with Calorimeters together with the results.
%In Section \ref{sec:tile_perf}  muon tagging with the Tile Calorimeter (TileCal) is presented.
%Finally, Sections \ref{sec:trig_from_data} and \ref{sec:highmass}  show the trigger performance on
%the di$-$muon final states $Z \rightarrow \mu\mu$ and $Z^\prime \rightarrow \mu\mu$  and explain how the trigger efficiency can be determined
%from real data.

\section{Detector simulation and data samples }
\label{Datasamples} The samples used  in this paper were produced
using a full GEANT4 based simulation of the ATLAS detector.
% produced
% within the Computing System Commissioning  (CSC) Data Challenge.
%The full list of CSC samples used in this note is shown in Appendix A.
%All the samples were simulated with ATLAS software release 12 and detector description version
%ATLAS-CSC-01-02-00. This geometry takes into account several realistic effects as
%chamber misalignment obtained by tilting and shifting randomly the muon chambers
%   with an RMS of 1 mrad and 1 mm respectively, and
%magnetic field map with initial displacements.
%Additional checks have been performed on the same samples reconstructed with the most recent reconstruction version implemented in
% {\scshape Athena} release 13.
%The analyses have been performed both from the Analysis Object Data (AOD) and from custom ROOT ntuples
%which are produced by running the muon slice over RDO files.
%L1 Simulation and HLT reconstruction were run with ATLAS release 12.0.6,
%using the same geometry database of simulation.
The trigger simulation options included both standard and
$B$-physics trigger simulation configurations, which correspond to
the standard  and the low trigger thresholds (see
Section~\ref{slice}). The deterioration of efficiency due to the
geometrical acceptance and the limited size of coincidence window
are taken into account in the L1 simulation and trigger logic
emulator.

Large samples of single prompt muons, simulated uniformly in
$\eta-\phi$, with fixed $\pt$ ranging from 2 GeV to 1 TeV, have been
used to study the muon trigger performance.
%A detailed list of the used datasets is shown in Table \ref{tab:single_muons}.
%The motivation of such large sample is two-fold: the need to have a reasonable estimation
%of the very low efficiency of the Muon Trigger system
%for low pt muons ($10^{-5}$ for trasverse momentum around 2 GeV)
%and the necessity to refine the L2 LUTs for some key $\pt$ threshold (6, 20 and 40 GeV).
One of the main backgrounds for the muon trigger selection comes
from in-flight decays of charged kaons and pions. This has been
evaluated using samples of minimum bias events and single pions,
where the mesons are forced to decay inside the Inner Detector
cavity in order to facilitate
the production of a sizable sample of  in-flight $\pi/\kaon$  decays.
%The list of the used datasets is shown in Table \ref{tab:single_pions}.
Muon trigger rates were determined using both single muons and minimum bias events.
% listed
%in Table \ref{tab:min_bias}.
The selection of muons using the Tile Calorimeter has been studied
using low $\pt$  (4 and 6 GeV) single muons
%(in Table \ref{tab:single_muons})
and semi-inclusive $b$ quark decays, $b\bar{b}\rightarrow\mu X$.
%listed in Table \ref{tab:bbtile}
%\noindent
%(??)
%The  accidental muon triggers coming from uncorrelated hits in the muon chambers (the muon cavern background)
%have been evaluated using sample of minimum bias events
% and single muons events with pile-up and muon cavern background corresponding
%to low luminosity running ({\cal L}=10$^{33}$ cm$^{-2}$ s$^{-1}$) and different levels of
%safety factors superimposed.
%(??)
Muon trigger studies on high-$p_{T}$ dimuon final states  and  the
determination of trigger efficiency from data have been performed
using
\Zmm
%$Z \rightarrow \mu\mu$
and $Z^\prime \rightarrow \mu\mu$ as
signal processes and $B \rightarrow \mu\mu$, W boson decays, $Z \rightarrow \tau \tau$ and  top-pair  events as background
processes.
%This analysis have been carried out using the AOD samples listed in
%Table \ref{tab:highpt}.

%All simulation,  digitization and reconstruction were done using the
%LCG GRID through Ganga~\cite{Ganga} and OSG GRID through Panda~{Panda}.
%Some  technical aspects of the analyses required the development of analysis tools  reading directly
%from raw data (RDO) and producing custom information (for expert use) in AANT together with standard AOD.
%To cope with the spirit of CSC and  full exercize the Distributed Analysis tools we sent 6k analysis jobs in LCG
%and  around 3k jobs in OSG.

\section{Muon trigger algorithms and configuration }
\label{slice}
%In this section the general configuration of the Muon Trigger is discussed.
%Since it is continously evolving, a version of the code has to be singled out
%as the reference for this note. For this purpose the version mostly used the studies reported
%in this work is chosen (ATLAS release 12.0.6).
%Readers  will need to do the required adaptions to newer versions of the muon trigger  slice.

%%% L1 DESCRIPTION
The L1 muon trigger selects active RoIs, in the event using
Resistive Plate Chambers (RPC) \cite{muon} in the barrel ($|\eta|<$
1.05) and Thin Gap Chambers (TGC) \cite{muon} in the endcaps (1.05
$<|\eta|<$ 2.4). The trigger algorithms look for hit coincidences
within different RPC or TGC detector layers inside the programmed
geometrical windows which define the transverse momentum region. A
coincidence is required in both $\eta$  and $\phi$ projections. The
information about muon candidates in both the barrel and the end-cap
is transmitted to the Muon to Central Trigger Processor Interface
(MuCTPI) \cite{muon}, which calculates the number of L1 muon
candidates for six different  \pt\  thresholds and takes overlaps
between the trigger sectors into account by using look-up-tables
(LUT).
%The L1 signatures, or trigger items, are combinations of requirements
%(or trigger conditions) on the multiplicities of various kinds of candidate objects delivered
%by the  muon triggers.
There are several L1 signatures each corresponding to a different
\pt\ threshold:
\begin{itemize}
\item mu0, mu5, mu6, mu8, mu10 for the low   \pt\ selection;
\item mu11, mu20, mu40  for the high   \pt\ selection.
\end{itemize}
\noindent The integer numbers after the mu'' symbolize the
required \pt\ threshold. L1  also provides the coordinates in $\eta$
and $\phi$ of the selected RoIs. The mu0 threshold represents a L1
configuration with completely open coincidence windows; it is also
called the Cosmic'' threshold as it can be used to trigger on
cosmic rays during the detector commissioning phase and between the
LHC fills. Similar thresholds, labeled with muXX'', have been
defined for L2 and EF.

The muon HLT runs L2 and EF algorithms. It starts from the RoI
delivered by the L1 trigger and applies trigger decisions in a
series of steps, each refining the existing measurement by acquiring
additional information from the ATLAS detectors. A list of physics
signatures, implemented in the event reconstruction and selection
algorithms, are used to build signature and sequence tables for all
HLT steps. This stepwise and seeded processing of events is
controlled by the trigger steering. The reconstruction progresses by
calling feature extraction algorithms. These typically request
detector data from within the RoI and attempt to identify muon
features.
%At the end algorithms update the RoI position if it has been more accurately determined.
Subsequently, a hypothesis algorithm determines whether the
identified feature meets the criteria necessary to continue. The
decision to reject the event or continue is based on the validity of
signatures, taking into account prescale and pass-through factors.
Thus,  events can be rejected after an intermediate step if no
signatures remain viable.

The main algorithm of the muon L2 system, muFast, runs on full
granularity data within the RoI defined by L1. An optimized strategy
is used to avoid heavy calculations and access to external services
to reduce the execution time of the algorithm. After pattern
recognition driven by the trigger hits which selects Monitored Drift
Tubes (MDT) regions crossed by the muon track, a track fit is
performed using MDT drift time precision measurements. The \pt
evaluation is performed using LUT.
%At L2 Inner Detector
Reconstructed tracks in the  Inner Detector
%\cite{sitrack,idscan}
can be combined with the tracks found
by muFast by a fast track combination algorithm called muComb.

The L2 algorithm (muIso) is used to discriminate between isolated and
non-isolated muon candidates by examining energy depositions in the
electromagnetic and hadronic calorimeters.
%The discrimination between  isolated and not-isolated muon candidates,
% by looking at differences in the energy patterns released in the electromagnetic and hadronic calorimeters,
%is done by L2 muIso algorithm.
The algorithm is seeded by muons selected by muFast or muComb and
decodes LAr and Tile Calorimeter quantities  in cones centered around the muon direction.
For the muon selection two different concentric cones are defined:
an internal cone chosen to contain the energy deposit
deposited by the muon itself, and
an external cone, containing energy only from detector noise, pile-up
and jet particles.

A strategy for tagging muons at L2 in the TileCal  is implemented
in the TileMuId algorithm. It can provide additional redundancy and robustness to the muon
trigger, as well as enhance the efficiency in the low $p_{\rm T}$ region.
%This algorithm exploits the full TileCal radial and transverse segmentation.
%The muon candidates are defined according to their energy deposition in the cells.
The search starts from the outermost calorimeter layer, which contains the cleanest signals, and once
a deposited energy is compatible with a muon, the algorithm checks the energy deposition
in the neighboring cells for the internal layers. Candidates are considered  tagged muons when
muon compatible  cells are found following a $\eta$-projective pattern in all the three TileCal layers.
There are two different variants of this algorithm :  one (TrigTileLookForMuAlg) is fully executed on the L2 Processing Unit (L2PU)
while the other  (TrigTileRODMuAlg) has a core part executed on the Readout Driver (ROD).
%~\cite{ROD}
%in order to save time.

%Three muon hypothesis algorithms have been implemented in ATLAS release 12.0.6 :  MufastHypo,
% MucombHypo,
% TrigMooreHypo.
%When requested by the trigger menu,  the L2 muon selection chain is completed by the
%more algorithms and hypothesis tests, as for example  muIsol and TileMuid.
% (see Sections \ref{muisol}
%and \ref{tilemuid})

%All muon chains use the same set of algorithms.
% For every L1 muon candidate passing the L1 selection the muFast algorithm is run first followed
%by its hypothesis algorithm, MufastHypo.
%Then tracks are reconstructed around the muon in the Inner Detector with the SiTrack \cite{sitrack}
%or IDSCAN \cite{idscan} algorithms.
%The tracks reconstructed separately by muFast and in the Inner Detector are combined in the muComb algorithm,
%followed by the hypothesis algorithm MucombHypo.
The  EF  accesses the full event with its full granularity. Given
the larger admissible latency, it is possible to adapt algorithms
developed for the off-line reconstruction to the on-line framework,
an approach that minimises algorithm development. The EF processing
starts by reconstructing tracks in the Muon Spectrometer around the
muons found by L2 and is done by three instances of the
%TrigMoore
EF algorithm;
%running MOORE
%\cite{moore} ,
%MuId StandAlone and MuId
%Combined.
% \cite{muid}.
the first instance reconstructs tracks inside the Muon Spectrometer,
starting with a search for regions of activity within the detector,
and subsequently performing pattern recognition and full track
fitting. The second step  extrapolates muon tracks to their
production point. Finally the information from the first two steps
is combined with the  reconstructed tracks from the Inner Detector.
% \cite{ipat,EFID}.

The hypothesis algorithms define a set of HLT trigger thresholds by
applying cuts on the $\pt$ of the muon candidate. The muon trigger
efficiency is defined as
\begin{equation}
\frac{\rm The~number~of~events~with~a~triggered~muon}{\rm The~number~of~events~with~a~muon}
\end{equation}
\noindent The effective trigger thresholds are obtained in such a
way that at the nominal threshold value the efficiency is 90\% of
the corresponding efficiency without cuts. For this reason effective
thresholds are slightly lower than nominal thresholds.

%The results presented in this paper have been obtained using
%two kind of HLT configurations : the standard muon trigger chain (CSC-06) and the
%special configuration for 900 GeV LHC operation (CSC-06-900GeV).
%In the standard configuration the trigger menu  included the signature $\mu$6{$\ell$},
%$\mu$6 and $\mu20$.  The signature $\mu$6$\ell$ implements threshold cuts  $p_T>$ 2 GeV both
%at L2 and EF hypothsis algorithms.
%The 900 GeV'' configuration  has a special L1 threshold set : mu6 and mu8 were
%replaced by mu0 (coincidence windows completely opened) and mu5. In this configuratio
%the HLT signatures were set to $\mu$0 (implementing threshold cuts as in $\mu$6$\ell$)
%and  $\mu$5 ($p_T>$5 GeV).

\section{L1 performance}
\label{sec:l1_perf}
\subsection{Barrel muon trigger performance}
\label{sec:rpc_perf}

As mentioned earlier,  studies of muon trigger performance were
conducted using simulated samples of single muon events generated
over a large \pt\ and angular range.
%\pt\ threshold in this note.
%The first step is the realization of efficiency curves
%for the low-\pt\ thresholds: mu6, mu8, mu10, and the high-\pt\: mu11, mu20, mu40.
L1 selection algorithms show an efficiency greater than 99\% for
muons with \pt\ above threshold. The overall acceptance
(82\% low-$\pt$, 78\% high-$\pt$)  is due exclusively to geometrical regions of
the Muon Spectrometer not covered by the RPC. Figure
\ref{fig:ineff_geo} shows the  inefficient regions corresponding to
the magnet support structures  ($-2.3 \leq \phi \leq -1.7$ and $-1.4 \leq \phi \leq 0.9$) and the spectrometer central crack at $\eta \sim 0$, not covered by RPCs. The overall loss in geometrical
acceptance due to the $\eta=0$ crack is approximately $7\%$ while
the loss due to the support structure is about $5\%$. Moreover
smaller inefficiency patterns are clearly visible which are due to
magnetic ribs in small trigger sectors. The  geometrical acceptance
effects are visible also in Fig.~\ref{fig:eff_vs_angoli} where the
L1 efficiency above threshold is shown with respect to $\eta$ and
$\phi$.
%In particular the efficiency gap in the right (left) plot is due to \textit{feet sectors} (magnet ribs).
\begin{figure}[!thb]
\begin{center}
\includegraphics[width=6.5cm,height=5.5cm]{fig/mappa_ineff_geo.eps}
\end{center}
\caption{L1 geometrical acceptance in the $\eta$-$\phi$ plane.}
\label{fig:ineff_geo}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.4\textwidth, height=5cm]{fig/eff_vs_eta_th1_pt75.eps}
\hspace{0.05\textwidth}
\includegraphics[width=0.4\textwidth, height=5cm]{fig/eff_vs_phi_th1_pt75.eps}
\end{center}
\caption{$\eta$ and $\phi$ dependence of barrel trigger efficiency for single muons with a $\pt$=75 GeV.}
\label{fig:eff_vs_angoli}
\end{figure}
Figure \ref{fig:eff_curves_std} shows turn-on curves for low-$\pt$
%(mu6, mu8 and mu10)
and high-$\pt$
%(mu11, mu20 and mu40)
thresholds; the efficiencies at plateau  and  effective thresholds
%\footnotemark\footnotetext{Here we define effective threshold (\pt\($\epsilon$=90$\%$)) as the \pt\ for which the efficiency reach the 90\% of the plateau value} and sharpness\footnotemark\footnotetext{sharpness = $\equiv$ \pt\($\epsilon$=90$\%$) - \pt\($\epsilon$=10$\%$)}
are summarized in Table \ref{tab:schema_curve_efficienza}.
\begin{table}[htb]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Threshold   &  Plateau Efficiency      &  Effective Threshold (GeV)    &  Sharpness(GeV)     \\
\hline
\hline
mu6         &  $0.82$             &   5.3                     &  2.2  \\
\hline
mu8         &  $0.82$             &   6.1                     &  1.9  \\
\hline
mu10        &  $0.82$             &   6.7                     &  2.2  \\
\hline
mu11        &  $0.78$             &   10.9                    &  3.7  \\
\hline
mu20        &  $0.78$             &   15.3                    &  7.1  \\
\hline
mu40        &  $0.78$             &   27.8                    &  19.7 \\
\hline
\end{tabular}
\end{center}
\caption{Plateau efficiencies, effective thresholds, and sharpness for L1 signatures.
Sharpness is defined as the difference of $\pt$ corresponding to
90$\%$ and 10$\%$ of the plateau efficiency. }
\label{tab:schema_curve_efficienza}
\end{table}
\begin{figure}[htb]
\centering
\includegraphics[width=0.4\textwidth, height=5cm]{fig/low_pt_std.eps}
\hspace{0.05\textwidth}
\includegraphics[width=0.4\textwidth, height=5cm]{fig/high_pt_std.eps}
\caption{L1 barrel efficiency as a function of $\pt$ for low-pt (left) and high-pt (right) thresholds.}
\label{fig:eff_curves_std}
\end{figure}
%\begin{figure}[htbp]
%    \begin{center}
%        \includegraphics[width=0.45\textwidth, height=6cm]{fig/std_mup_mum/eff_th1_mup_mum.eps}
%   \hspace{0.05\textwidth}
%   \includegraphics[width=0.45\textwidth, height=6cm]{fig/asim/asim_th1_ing.eps}
%    \end{center}
%    \caption{Efficiency curves for  6 GeV threshold.}
%    \label{fig:asimmetry}
%\end{figure}

Muon tracks are deflected in the $r-\eta$ plane  under the action of
the toroidal magnetic field. Their trajectories are symmetrical
under charge exchange and reflection with respect to the plane
$z=0$, but the layout of the Muon Spectrometer is not. This
asymmetry,
%that here we define as $\alpha = \frac{\varepsilon^+ - \varepsilon^-}{\varepsilon^+ + \varepsilon^-}$
could, in principle, produce a bias in the trigger efficiency
calculation. From the single muon data sample, it was found that for
muons with $\pt$ greater than the L1 threshold the asymmetry in the
efficiency is quite small ($<1$\%).

% and is considerable only
%for muons with $p_T^\mu < p_T^{Thr}$.
%This samples will be try to HLT, and almost all rejected.
%Figure \ref{fig:asimmetry} shows the
%efficiency curves for $\mu^+$ and $\mu^-$, and their asymmetry respectively.

Particular attention was devoted to the study of muons with  $2 < \pt < 3.5$ GeV in the barrel region ($|\eta|$  $<$1.05). Given the
large inclusive cross-section with muons in the final state, this
very low-\pt\ region represents the most significant contribution to
the total expected muon rate.  Table \ref{tab:very_low_barrel} shows
the fraction of these events that produce hits in the RPCs and the
L1 barrel efficiency for passing the mu6 trigger. Most of the low
\pt muons that pass L1 have $|\eta|$ $\simeq$1.
\begin{table}[htb!]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Muon $\pt$ (GeV)   &  Percentage of events with hits in RPC   &  L1 Efficiency  \\
\hline
\hline
2          &  $0.15\%$       &   $(1.4\pm0.1)\cdot10^{-3}$ \\
\hline
2.5        &  $0.35\%$       &   $(3.4\pm0.1)\cdot10^{-3}$  \\
\hline
3          &  $0.48\%$       &   $(4.8\pm0.1)\cdot10^{-3}$  \\
\hline
3.5        &  $3.36\%$       &   $(33.4\pm0.3)\cdot10^{-3}$  \\
\hline
\end{tabular}
\end{center}
\caption{L1 RPC efficiency for very low-$\pt$ single muon events with $|\eta|$  $<$1.05.}
\label{tab:very_low_barrel}
\end{table}

%In past studies, for some technical reasons, barrel (RPC) and endcap region (TGC) have been
%defined using, for single muons sample, the value of $\eta$ at the interaction point.
%This is quite true for $\pt >6$ GeV.
Very low-\pt\  muons produced in the acceptance of the TGC
($|\eta|>$1.05) sometimes give hits in the RPC ($|\eta|<$1.05)
because they are strongly deflected by the  magnetic field. The
fraction of muons with $|\eta|>$1.05 that give RPC hits ranges from
46\% at $\pt=2$ GeV to about 10\% at 3.5 GeV, and is negligible
above 4 GeV.  The overall efficiency for such muons to pass the mu6
trigger is about $10^{-3}$.

Figure~\ref{fig:tgc_efficiency} shows L1 end-cap efficiency curves
for low~\pt\  (left) and  high~\pt\ thresholds (right). Efficiencies
at the threshold and plateau are summarized in
Table~\ref{tab:tgc_efficiency}. The efficiency of mu6 at threshold
is 77\%, relatively lower than other cases. This is due to the
limited window-size of the three-station coincidence for muons
having $\pt$ of 6 GeV.
\begin{figure}[thb!]
\begin{center}
\includegraphics[width=0.48\figwidth]{fig/tgc_effCurveLow.eps}
\includegraphics[width=0.48\figwidth]{fig/tgc_effCurveHi.eps}
\end{center}
\caption{The end-cap trigger efficiency curves for each \pt\ threshold.
The left plot shows the low-\pt\ thresholds of 6,~8, and 10 GeV and the right plot shows the high-\pt\ thresholds of 11,~20 and 40 GeV.}
\label{fig:tgc_efficiency}
\end{figure}
\begin{table}[thb!]
\begin{center}
\begin{tabular}{|l|c|c|c||c|c|c|}\hline
\pt\ threshold (GeV) & 6 & 8  & 10  & 11  & 20  & 40  \\ \hline
Threshold       & $77\%$ & $84\%$ & $88\%$ &  $88\%$ & $92\%$ & $90\%$ \\
Plateau         & $95\%$ & $95\%$ & $95\%$ &  $95\%$ & $94\%$ & $93\%$ \\ \hline
\end{tabular}
\caption{Trigger efficiencies at threshold and plateau for various muon \pt\ thresholds.
\label{tab:tgc_efficiency}}
\end{center}
\end{table}
The $\eta$ dependence of the mu6 and mu20 efficiency are shown in
Fig.~\ref{fig:tgc_efficiencyEtaThr}~(at threshold) and
Fig.~\ref{fig:tgc_efficiencyEtaPla}~(at plateau) with respect to the
sign of charge of muon $q\times\eta$. Because the two muon end-cap
stations are made as mirror images, $\mu^{-(+)}$ with $\eta>(<)~0$
behaves the same as $\mu^{+(-)}$ with $\eta<(>)~0$. The difference
of efficiency between the two signs of $q\times\eta$ is large at the
geometrical boundary for muons near the $\pt=6$ threshold  as shown
in Fig.~\ref{fig:tgc_efficiencyEtaThr}. One more point worth noting
is the dip at $\eta$=2 for the mu6 signature. Muons in the dip
region pass through chambers which belong to different trigger
sectors, consequently the requirement of a three-station coincidence
is not satisfied and trigger efficiency is reduced.
Figure~\ref{fig:tgc_efficiencyPhiPla} shows the $\phi$ dependence of
mu6 (left) and mu20 (right) trigger efficiencies at the plateau and
at threshold. The effect of octant symmetry of the magnetic field is
seen in the plot of the mu6 efficiency for threshold muons. For the
mu6 signature at plateau and for the mu20 signature, this effect is
not observed, resulting in an approximately uniform efficiency.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.48\figwidth]{fig/tgc_effEtaThrLow.eps}~
\includegraphics[width=0.48\figwidth]{fig/tgc_effEtaThrHi.eps}
\end{center}
\caption{$\eta$ dependence of end-cap trigger efficiency for mu6 (left)
and mu20 (right).
The solid circles represent $q\times\eta>0$, the open circles represent $q\times\eta<0$.}
\label{fig:tgc_efficiencyEtaThr}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.48\figwidth]{fig/tgc_effEtaPlaLow.eps}~
\includegraphics[width=0.48\figwidth]{fig/tgc_effEtaPlaHi.eps}
\end{center}
\caption{$\eta$ dependency of end-cap trigger efficiency at plateau for mu6 (left) and mu20 (right).
The solid circles represent $q\times\eta>0$, the open circles represent $q\times\eta<0$.}
\label{fig:tgc_efficiencyEtaPla}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.48\figwidth]{fig/tgc_effPhiLow.eps}~
\includegraphics[width=0.48\figwidth]{fig/tgc_effPhiHi.eps}
\end{center}
\caption{$\phi$ dependence of end-cap trigger efficiency for mu6 (left) and mu20 (right) for muons with $\pt = 45$ GeV.
The open circles show the efficiency at threshold and the solid circles show the efficiency at plateau.}
\label{fig:tgc_efficiencyPhiPla}
\end{figure}

\section{Performance of L2 muon algorithms}
\label{sec:l2_perf}
As described in  Section \ref{Datasamples}, algorithm performance is
evaluated on samples of single muons generated with different
transverse momenta.
%The trigger simulation version is 12.0.6.
The resolution of the inverse of the measured momentum with respect
to the generated transverse momentum is studied. Due to the
non-uniform magnetic field in the Muon Spectrometer it is divided
into four regions  according to the pseudorapidity of the muon
candidate:  the Barrel region with $|\eta|<1.05$, and three  end-cap
regions with
$1.05<|\eta|<1.5$, $1.5<|\eta|<2.0$,  and
$2.0<|\eta|<2.4$.

% \begin{figure}[htbp]
%   \begin{center}
%     \includegraphics[width=0.3\columnwidth]{fig/ms_magnetic_field.eps}
%   \end{center}
%   \caption{Magnetic field integral along a straight trajectory ($\oint % B.dl$) as a function of the trajectory's pseudorapidity
%   \label{fig:ms_magnetic_field}}
% \end{figure}
%To take care of the non-gaussianity of the momentum resolution, the
%effective momentum thresholds are choosen as
% the ones that give $90\%$ efficiency for the nominal threshold. For
%example, the effective threshold correponding to $mu20$ is the
%momentum threshold such that $90\%$ of muons with 20 GeV are
%reconstructed with a momentum above the effective
%threshold\footnote{The effective threhold for $mu4$ is the only
%exception to this definition. Having in ming the ATLAS b-physics
%programme, a  lower threshold has been choosen.}.

%\subsection{muFast}
For the Muon Spectrometer standalone reconstruction (muFast), the
resolution of inverse $\pt$ as a function of the muon transverse
momentum is shown in Fig.~\ref{fig:mufast_muon_resolution}(left).
The degradation in the resolution with respect to previous results
\cite{pisa_meet} is caused by the realistic geometry misalignment
introduced in the muon simulation. Resolution as function of  $\eta$
and $\phi_{Loc}$\footnote{$\phi_{Loc}$ is the azimuthal angle folded
up
in $[0,\pi/16]$ such to cover half of an odd MS sector ($[0,\pi/32]$) and half of
an even MS sector $[\pi/32,\pi/16]$.} is shown in Fig.~\ref{fig:mufast_muon_resolution}(right).
The degradation of the
resolution in the endcap regions is evident.
%The resolution as a function of
%$\eta$ and $\phi$ is shown in
%Fig. \ref{fig:mufast_muon_resolution_eta_phi}
%for a $40$\, GeV/$c$
%muon.
%The
%small asymmetry of the resolution in pseudorapidity observed is caused by the fact that
%we used for this study only positively charged muons.
%The different regions in
%$\eta \times \phi$ where the momentum resolution is homogenous are
%shown in Figure \ref{fig:mufast_muon_resolution-b}.
%With the  momentum resolutions evaluated, the effective thresholds are
%shown in Table \ref{tab:lvl2_mufast_threshold}.
% \begin{figure}[htbp]
% \centering
% \begin{minipage}[c]{0.45\textwidth}
% \includegraphics[width=0.9\columnwidth]{fig/mufast_muon_resolution.eps}
% \end{minipage}
% \begin{minipage}[c]{0.45\textwidth}
% \includegraphics[width=0.9\textwidth]{fig/resolution_mufast_eta_phi_2.eps}
% \end{minipage}
% \caption{Momentum resolution of muFast as a function of momentum (left).  Momentum resolution of muFast as a function of
% pseudorapidity and azimuthal
% angle $\phi$ for a 40 GeV muon. (right)
% \label{fig:mufast_muon_resolution}.}
% \end{figure}
\begin{figure}[htb]
\begin{center}
%\subfigure[Resolution {\em vs} $P_{T}$]{\label{fig:mufast_muon_resolution-a}
%\includegraphics[width=0.48\columnwidth]{fig/mufast_muon_resolution.eps}}
%\subfigure[Resolution {\em vs} $\eta$ and $\phi$ ($P_T=40$GeV)]{\label{fig:mufast_muon_resolution-b}
%\includegraphics[width=0.48\textwidth]{fig/resolution_mufast_eta_phi_2_bw.eps}}
\subfigure{\includegraphics[width=0.48\columnwidth]{fig/mufast_muon_resolution.eps}}
\subfigure{\includegraphics[width=0.48\columnwidth]{fig/resolution_mufast_eta_phi_2_bw.eps}}
\end{center}
\caption{1/$\pt$ resolution (Muon Spectrometer StandAlone) as a
function of $\pt$ (left) and
$\eta-\phi_{Loc}$ (right).}
\label{fig:mufast_muon_resolution}
\end{figure}
%\begin{figure}[htb]
%\begin{center}
%\subfigure[Resolution {\em vs} $\eta$]{\label{fig:mufast_muon_resolution_eta_phi-a}\includegraphics[width=0.45\columnwidth]{fig/mufast_muon_resolution_eta.eps}}
%\subfigure[Resolution {\em vs} $\phi_{loc}$ ($P_T=40$GeV)]{\label{fig:mufast_muon_resolution_eta_phi-b}\includegraphics[width=0.45\textwidth]{fig/mufast_muon_resolution_phi.eps}}
%\end{center}
%\caption{Muon Momentum resolution (StandAlone)$P_T=40$GeV}
%\label{fig:mufast_muon_resolution_eta_phi}
%\end{figure}
%\begin{table}
%  \begin{center}
%    \caption{Effective momentum thresholds evaluated for muFast for
%different pseudorapidity regions.
%    \label{tab:lvl2_mufast_threshold}}
%    \begin{tabular}{|l|c|c|c|c|}\hline
%      Nominal threshold      & $\eta|<1.05$ & $1.05<|\eta|<1.5$ &
%$1.5<|\eta|<2.0$ & $2.0<|\eta|<2.4$ \\
%      \hline \hline
%mu4  &  3.0  &    2.5 &    2.5 &   2.5 \\
%mu5  &  4.6  &    3.3 &    4.0 &   4.5 \\
%mu6  &  5.4  &    4.5 &    4.9 &   5.3 \\
%mu8  &  7.2  &    6.7 &    6.4 &   7.3 \\
%mu10 &  8.9  &    9.0 &    8.4 &   9.2 \\
%mu11 &  9.8  &   10.1 &    9.3 &  10.1 \\
%mu15 &  13.0 &   14.0 &   13.0 &  14.0 \\
%mu20 & 17.5  &   18.5 &   17.0 &  18.0 \\
%mu40 & 31.5  &   30.0 &   28.5 &  32.5 \\
%\hline
%    \end{tabular}
%  \end{center}
%\end{table}
The muFast efficiency with respect to L1 selection as a function of
muon momentum are shown for mu4, mu6, and mu20 selections in
Fig.~\ref{fig:lvl2_mufast_turnon}. The low rejection at small
momentum ($2.5 < \pt < 4.5$\, GeV), in particular in the barrel
region, is caused by candidate tracks not pointing to the nominal
interaction vertex due to large scattering angles.
% \begin{figure}[htbp]
% \begin{center}
% \subfigure[{\em muFast} efficiency with respect to L1 selection
% for mu4, mu6 and mu20 trigger selection for
% different pseudorapdity regions]{\label{fig:lvl2_mufast_turnon}
% \includegraphics[width=0.40\columnwidth]{fig/lvl2_mufast_eff_2.eps}}
% \subfigure[Event display of a low momentum muon (\pt = 2 GeV) passing the mu20
% selection (both L1 and muFast). The missed segment in the inner
% station is clearly visible]{\label{fig:mufast_event_display}\includegraphics[width=0.40\textwidth]{fig/lvl2_mufast_eventdisplay.eps}}
% \end{center}
% %\caption{Muon Momentum resolution (StandAlone)}
% %\label{fig:mufast_muon_resolution}
% \end{figure}
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=0.85\figwidth]{fig/lvl2_mufast_eff_3.eps}
\end{center}
\caption{ muFast efficiency with respect to L1 selection
for the mu4, mu6, and mu20 triggers for different $\eta$ regions.
\label{fig:lvl2_mufast_turnon}}
\end{figure}

% \begin{figure}[htbp]
%   \begin{center}
%     \includegraphics[width=0.5\columnwidth]{fig/lvl2_mufast_eventdisplay.eps}
%   \end{center}
%   \caption{Event display of a low momentum muon (\pt = 2 GeV) passing the mu20
% selection (both L1 and muFast). The missed segment in the inner
% station is clearly visible
%   \label{fig:mufast_event_display}.}
% \end{figure}

Efficiencies of the MS standalone reconstruction (muFast) (mu6
trigger selection) with respect to L1 selection as a function of
$\eta$ and $\phi_{Loc}$ for muons with $P_T=6$\, GeV are shown in
Fig.~\ref{fig:mufast_muon_mu6_eff_eta_phi}.

\begin{figure}[htb!]
\begin{center}
%\subfigure[Efficiency {\em vs} $\eta$ ($P_T=6$GeV) ]{\label{fig:mufast_muon_mu6_eff-a}
%\includegraphics[width=0.48\columnwidth]{fig/mufast_eff_eta_mu6.eps}}
%\subfigure[Efficiency {\em vs} $\phi$ ($P_T=6$GeV)]{\label{fig:mufast_muon_mu6_eff-b}
%\includegraphics[width=0.48\textwidth]{fig/mufast_eff_phi_mu6.eps}}
\subfigure{\includegraphics[width=0.46\columnwidth]{fig/mufast_eff_eta_mu6.eps}}
\subfigure{\includegraphics[width=0.46\textwidth]{fig/mufast_eff_phi_mu6.eps}}
\end{center}
\caption{Efficiency of muFast as a function of $\eta$ (right) and $\phi_{Loc}$ (left)
for muons with $\pt=6$ GeV.}
\label{fig:mufast_muon_mu6_eff_eta_phi}
\end{figure}

%\subsection{muComb}
%A further step in the trigger reconstruction at L2 is performed
%by
The combination of a Muon Spectrometer standalone muon candidate
with an Inner Detector  track found by the L2 tracking algorithms is
performed by muComb. For muons with $\pt < 50$ GeV the Inner
Detector measurement has a better resolution than the Muon
Spectrometer standalone measurement.
Therefore,
the combination of the two measurements gives better resolutions in the low-$\pt$
range.
%agb commented because latex is screwy and wants to call this figure 5!!!
%Figure~\ref{fig:mucomb_mures}
Figure 11 shows the  1/$\pt$
resolution as a function of $\pt$ and also the resolution as
function of $\eta$ and $\phi_{Loc}$.
% The resolution as a function of
%pseudorapidity and azimuthal angle are shown respectively in
%Figure \ref{fig:mucomb_muon_resolution-b}.
%As expected, the effective momentum
%thresholds are larger than those for muFast.
%(Tab.\ref{tab:lvl2_mucomb_threshold})
The muComb efficiency with respect to muFast as a function of $\pt$
for mu4, mu6, and mu20 is shown in
Fig.~\ref{fig:lvl2_mucomb_turnon}. The problem of low rejection for
low-$\pt$ muons is partially solved when the Muon Spectrometer
candidates are combined with Inner Detector tracks.
\begin{figure}[htb]
\label{fig:mucomb_mures}
\begin{center}
%\subfigure[Momentum resolution of muComb {\em vs}
%$P_T$]{\label{fig:mucomb_muon_resolution-a}\includegraphics[width=0.48\columnwidth]{fig/mucomb_muon_resolution.eps}}
%\subfigure[Momentum resolution of muComb {\em vs}
%$\eta$ and $\phi$]{\label{fig:mucomb_muon_resolution-b}
%\includegraphics[width=0.48\columnwidth]{fig/resolution_mucomb_eta_phi_2_bw.eps}}
\subfigure{{\includegraphics[width=0.48\columnwidth]{fig/mucomb_muon_resolution.eps}}}
\subfigure{{\includegraphics[width=0.48\columnwidth]{fig/resolution_mucomb_eta_phi_2_bw.eps}}}
\caption{The muon combined 1/$\pt$
resolution as a function of $\pt$ (left) and as a function of
$\eta-\phi_{Loc}$ (right).}
\end{center}

\end{figure}
%\begin{figure}[htb]
%\centering
%\subfigure[Momentum resolution of muComb {\em vs} $\eta$]{\label{fig:mucomb_muon_resolution_eta_phi-eta}
%\includegraphics[width=0.45\textwidth]{fig/mucomb_muon_resolution_eta.eps}}
%\subfigure[Momentum resolution of muComb {\em vs} $\phi$]{\label{fig:mucomb_muon_resolution_eta_phi-phi}
%\includegraphics[width=0.45\textwidth]{fig/mucomb_muon_resolution_phi.eps}}
%\caption{Momentum resolution of muComb ($P_T=40GeV$)}
%\label{fig:mucomb_muon_resolution_eta_phi}
%\end{figure}
%\begin{table}
%  \begin{center}
%    \caption{Effective momentum thresholds evaluated for muComb for
%different pseudorapidity regions.
%    \label{tab:lvl2_mucomb_threshold}}
%    \begin{tabular}{|l|c|c|c|c|}\hline
%      Nominal threshold      & $\eta|<1.05$ & $1.05<|\eta|<1.5$ &
%$1.5<|\eta|<2.0$ & $2.0<|\eta|<2.4$ \\
%      \hline \hline%

%mu4  &     3.0  &    2.5  &     2.5  &    2.5 \\
%mu5  &     4.9  &    4.8  &     4.8  &   4.8 \\
%mu6  &     5.8  &    5.8  &     5.8  &   5.6 \\
%mu8  &     7.8  &    7.7  &     7.7  &   7.7 \\
%mu10 &     9.8  &    9.5  &     9.6  &   9.7 \\
%mu11 &    10.8  &   10.4  &    10.6  &  10.6 \\
%mu15 &    14.5  &   14.0  &    14.0  &  14.5 \\
%mu20 &    19.5  &   18.5  &    18.5  &  18.5 \\
%mu40 &    37.5  &   37.0  &    37.0  &  35.0 \\ \hline
%    \end{tabular}
%  \end{center}
%\end{table}
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=0.85\figwidth]{fig/lvl2_mucomb_wrt_mufast_eff_3.eps}
\end{center}
\caption{The muComb algorithm efficiency with respect to mu4, mu6, and mu20 triggers  for the
different $\eta$ regions.
\label{fig:lvl2_mucomb_turnon}}
\end{figure}

Efficiencies of muComb (mu6 trigger selection) with respect to
muFast selection as a function of $\eta$ and $\phi_{Loc}$  for muons
with $\pt=6$\, GeV are shown in
Fig.~\ref{fig:mucomb_muon_mu6_eff_eta_phi}.

\begin{figure}[htb!]
\begin{center}
%\subfigure[Efficiency {\em vs} $\eta$ ($P_T=6$GeV) ]{\label{fig:mucomb_muon_mu6_eff-a}
%\includegraphics[width=0.48\columnwidth]{fig/mucomb_eff_eta_mu6.eps}}
%\subfigure[Efficiency {\em vs} $\phi$ ($P_T=6$GeV)]{\label{fig:mucomb_muon_mu6_eff-b}
%\includegraphics[width=0.48\textwidth]{fig/mucomb_eff_phi_mu6.eps}}
\subfigure{\includegraphics[width=0.46\columnwidth]{fig/mucomb_eff_eta_mu6.eps}}
\subfigure{\includegraphics[width=0.46\textwidth]{fig/mucomb_eff_phi_mu6.eps}}
\end{center}
\caption{Efficiency of muComb as a function of $\eta$ (right) and
$\phi_{Loc}$ (left) for single muons with $\pt=6$ GeV.}
\label{fig:mucomb_muon_mu6_eff_eta_phi}
\end{figure}

%Single muons with transverse momentum at the vertex between 3 GeV and 1 TeV and with simulated vertex spread (0.015, 0.015, 56.) mm have been analized.
\section{Event filter performance}
\label{sec:ef_perf} The full reconstruction  in the Muon EF has been
executed on the simulated samples described in
Section~\ref{Datasamples}. The reconstruction in the Muon
Spectrometer is carried out by the MOORE algorithm, the
extrapolation to the vertex of the muon track found in the Muon
Spectrometer is performed by the MuId standalone algorithm and the
combination of the tracks found in the Muon Spectrometer and in the
Inner Detector by the MuId Combined algorithm.

Efficiency for single muon events is defined as the ratio of events with a reconstructed track at the EF
after the execution of each reconstruction step
%and  passing a loose quality cut on $\chi^2$/d.o.f. from MOORE ($\chi^2 < 3$)
to all events which have passed  L1 and L2. The efficiency with
respect to L2 as a function of muon \pt\ is shown in
Fig.~\ref{EF:efficiency_vs_pt} for all three EF algorithms.
\begin{figure}[h!]\begin{center}
\includegraphics[width=0.6\figwidth]{fig/EF_efficiency.ps}
\caption{Efficiency as a function of muon $\pt$ for MOORE, MuId Standalone and MuId Combined.\label{EF:efficiency_vs_pt}}
\end{center}\end{figure}
The efficiency is defined on an event basis, and counts only once
events having L2 muon-feature or EF track multiplicity greater than
1. According to this definition, the efficiency to trigger an event
with more than one muon is expected to be higher with respect to
what is estimated here.

The efficiencies are lower for $3 < \pt <6$  due to multiple
scattering and energy loss fluctuation effects. Moreover, in the
case of MuId Combined, at very high $\pt$ the increasing probability
of muon showering is responsible for a small loss in efficiency. The
efficiencies as a function of $\eta$ and $\phi$ show a structure,
especially at low momentum, explainable with some residual
dependence in  $\eta$ and $\phi$ on the Muon Spectrometer
geometrical acceptance and on the magnetic field inhomogeneities
which affect less previous levels.
% explainable with a  dependence
%on the muon spectrometer geometrical acceptance both in $\eta$ and $\phi$.
It can be seen in Fig.~\ref{EF:eff_vs_eta} where the efficiency is
shown for 6 GeV muons.
\begin{figure}[htb!]\begin{center}
\includegraphics[width=0.48\figwidth]{fig/EF_eff_vs_eta_6.ps}
\includegraphics[width=0.48\figwidth]{fig/EF_eff_vs_phi_6.ps}
\caption{Efficiency of MuId Combined as a function of $\eta$ (left)
and of $\phi$ (right) for 6 GeV muons.\label{EF:eff_vs_eta}}
\end{center}\end{figure}
In Fig.~\ref{EF:eff_pt} the MuId combined efficiency with respect to
L2 for different thresholds is shown on the left, while the overall
trigger efficiency (L1 + L2 + EF) with respect to the generated
muons is shown on the right. All efficiency values are averaged over
the whole  $|\eta|<$2.4 range.

%In Section \ref{EF:rates} the efficiency of MuId combined versus \pt\ will be shown as a function of the \pt\ thresholds
%applied in the TrigMoore hypothesis algorithms.
%Efficiency is defined in this case as the ratio of events with a reconstructed track from MuId combined exceeding a given
%\pt\ threshold to all events with a muon passing the L2 trigger.
%\subsection{Resolutions}
\begin{figure}[htb!]\begin{center}
\includegraphics[width=0.48\figwidth]{fig/EF_eff_pt.ps}
\includegraphics[width=0.48\figwidth]{fig/GlobalEfficiencies.ps}
\caption{ MuId combined efficiencies for various \pt\ thresholds
with respect to L2 (Left) and with respect to generated truth muons
(Right).
% $p_T = 4,5,6,8,10,11,15,20,40$ GeV.
\label{EF:eff_pt}}
\end{center}\end{figure}

%The $1/p_T$ resolution for 6 GeV muons reconstructed in the EF with MuId Combined is shown in Fig.~\ref{EF:ptres}
%for the barrel and the endcaps regions.
%According to a Gaussian fit to the resolutions,
%the distributions are  centered around 1 and do not have non Gaussian tails.
%\begin{figure}[htb!]\begin{center}
%\includegraphics[width=0.48\figwidth]{fig/EF_MuIdcb_6_bar.ps}
%\includegraphics[width=0.48\figwidth]{fig/EF_MuIdcb_6_ec.ps}
%\caption{MuId combined $1/p_T$ resolution for 6 GeV muons in the barrel (left)
%and in the endcap (right) regions.\label{EF:ptres}}
%\end{center}\end{figure}
In Fig.~\ref{EF:ptres_vs_pt}, the 1/$\pt$ resolution  is shown as a
function of muon $\pt$ for all EF algorithms. For a muon with $\pt$
below 50 GeV the Inner Detector dominates the reconstruction
precision so the combination of measurements greatly improves the
resolutions. For a muon with  $\pt$ above 100 GeV, the Muon System
dominates the measurement of the muon combined transverse momentum.
\begin{figure}[h!]\begin{center}
\includegraphics[width=0.65\figwidth]{fig/EF_ptres_vs_pt.ps}
\caption{1/$\pt$ resolution as a function of $\pt$ for MOORE, MuId Standalone and MuId Combined.\label{EF:ptres_vs_pt}}
\end{center}\end{figure}
%Results on $\phi$ and $\eta$ resolutions are
%reported in Fig.~\ref{EF:etaphires} as obtained with Moore, MuId standalone and MuId combined as a function of the muon transverse momentum. As expected, these resolutions
%deteriorate at low \pt\, owing to the multiple scattering effect.
%Remarkable improvements by MuId combined are evident with respect to the standalone muon reconstruction (up to two orders of magnitude).
%\begin{figure}[h!]\begin{center}
%\includegraphics[width=0.48\figwidth]{fig/EF_phires_vs_pt.ps}
%\includegraphics[width=0.48\figwidth]{fig/EF_etares_vs_pt.ps}
%\caption{Resolution (in rad) on azimuthal angle $\phi$  (left) and pseudorapidity $\eta$ (right) as a function
%of muon \pt\ in the cases of Moore, MuId standalone and MuId combined. All values are averaged over the full $\eta$ range.\label{EF:etaphires}}
%\end{center}\end{figure}
1/$pt$ resolution as a function of $\eta$ is shown in
Fig.~\ref{EF:ptreso_vs_eta} for 20 GeV muons. The worsening of the
resolution in the region 1.0 $<|\eta|<$ 1.5 can be attributed to the
highly inhomogeneous magnetic field in the transition regions of the
Muon Spectrometer.
This effect is recovered by means of the combined reconstruction which exploits the Inner Detector performance.
\begin{figure}[h!]\begin{center}
\includegraphics[width=0.6\figwidth]{fig/EF_ptreso_vs_eta.ps}
\caption{$1/p_T$ resolution for MuId Combined  as a function of
$\eta$ for muons with $p_T =$ 20 GeV. \label{EF:ptreso_vs_eta}}
\end{center}\end{figure}
%As an outcome of the fact that the muon EF algorithms are the same used in the offline reconstruction,
%performance results
%shown so far in all the acceptance regions are in good agreement with those found in the offline tracking environment.
%\subsection{Muon EF rates \label{EF:rates} }
%The trigger rates for single muons coming from all relevant physical processes expected in ATLAS have been computed running the full muon vertical slice.
%Different transverse momentum thresholds are considered depending on the various luminosity scenarios foreseen for data taking: from  L = $10^{31} cm^{-2} s^{-1}$
%to L = $10^{34} cm^{-2} s^{-1}$.
%Results shown here are obtained both for barrel and endcaps starting from {\bf muComb}  output at L2.

%The efficiency curves are the results of fits to a Fermi function and, to obtain them,
%the \pt\ estimate given by MuId combined has been used and the cut on \pt\ has been tuned
%in such a way that at the nominal threshold value the efficiency is 90\%
%of the corresponding one without cuts (i.e. Fig.~\ref{EF:efficiency_vs_pt}).
%Folding these efficiency curves with the known cross-sections for the relevant single muon processes
%(a 0.5 GeV step has been applied in the numerical integration), taking into account the dependence on $\eta$, the rates in Table \ref{EF:rates} are obtained for some low and high \pt\ thresholds in the barrel and in the endcaps, while in Fig.~\ref{EF:plot_rate}
%the total (barrel+endcaps) EF rates at L = $10^{31}$ cm$^{-2}$ s$^{-1}$ are reported as a function of the \pt\ threshold.

%\begin{table}
%\begin{center}
%\begin{tabular}{|c|c|c|c|c|}
%\hline
%\hline
%L = $10^{33}$ & 6 GeV & 6 GeV & 8 GeV & 8 GeV \\
%$cm^{-2} s^{-1}$ & Barrel (Hz) & Endcaps (Hz) & Barrel (Hz) & Endcaps (Hz) \\ \hline
%$\pi/K$ & 1900 & 1200 & 290 & 260 \\ \hline
%beauty & 1900 & 2200 & 550 & 800 \\ \hline
%charm & 2400 & 2800 & 640 & 930 \\ \hline
%top & $-$ & $-$ & $-$ & $-$ \\ \hline
%$W$ & 3 & 4 & 3 & 4 \\ \hline
%TOTAL & 6200 & 6200 & 1480 & 1990 \\ \hline
%\end{tabular}
%\\
%\begin{tabular}{|c|c|c|c|c|}
%\hline
%\hline
%L = $10^{34}$ & 20 GeV & 20 GeV & 40 GeV & 40 GeV \\
%$cm^{-2} s^{-1}$ & Barrel (Hz) & Endcaps (Hz) & Barrel (Hz) & Endcaps (Hz) \\ \hline
%$\pi/K$ & 50 & 40 & 0.1 & 0.2 \\ \hline
%beauty & 220 & 380 & 10.5 & 16.3 \\ \hline
%charm & 260 & 330 & 7.1 & 11.1 \\ \hline
%top & $-$ & $-$ & 0.1 & 0.1 \\ \hline
%$W$ & 20 & 30 & 3.9 & 6.1 \\ \hline
%TOTAL & 550 & 780 & 21.7 & 33.8 \\ \hline

%\end{tabular}
%\end{center}\vglue3mm
%\caption{Single muon trigger rates as obtained at EF with MuId combined, for different low and high \pt\ thresholds, respectively at L = $10^{33} cm^{-2} s^{-1}$ and L = $10^{34} cm^{-2} s^{-1}$. \label{EF:rates}}
%\end{table}

%\begin{figure}[h!]\begin{center}
%\includegraphics[width=0.65\figwidth]{fig/EF_muonrates.ps}
%\caption{Expected EF rates at L = $10^{31}$ cm$^{-2}$ s$^{-1}$ for single muon processes as functions of muon \pt\ threshold integrated over $\eta$ < $2.4$.
%\label{EF:plot_rate}}
%\end{center}\end{figure}

%A large contribution to the total rate is due to in-flight decays of light mesons, especially for what concerns the low-\pt\ thresholds. A strategy exploiting refined matching requirements between the tracks in the inner detector and the muon spectrometer has been developed to evaluate the reduction of such contribution to the final rate (see paragraph \ref{EF_pi_K}).

%As far as heavy flavour decays are concerned, their contributions have been evaluated by using a rather conservative approach, according to the cross-sections of muons from b and c decays assumed in the event generation with Pythia 6.4 \cite{pythia6}. The main source of uncertainty on heavy flavour rates come from the poor knowledge of these cross-sections.

\section{Muon trigger rates }
\label{sec:rates}
\begin{table}[t!]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multicolumn{7}{|c|}{L1 muon trigger rates}\\\hline\hline
& \multicolumn{2}{|c|} {${\cal{L}} = 10^{31}$ cm$^{-2}$ s$^{-1}$} & \multicolumn{2}{|c|} {${\cal{L}} = 10^{33}$ cm$^{-2}$ s$^{-1}$}  & \multicolumn{2}{|c|} {${\cal{L}} = 10^{34}$ cm$^{-2}$ s$^{-1}$} \\
\hline
& Barrel (Hz) & Endcaps (Hz) & Barrel (Hz) & Endcaps (Hz) & Barrel (Hz) & Endcaps (Hz)  \\
\hline
& \multicolumn{2}{|c|} {Cosmic''} & \multicolumn{2}{|c|} {6 GeV}  & \multicolumn{2}{|c|} {20 GeV} \\
%\hline
$\pi/K$ & 454 & 199     & 8600 & 5300    & 1100 &5200  \\ \hline
beauty  & 85  & 74      & 4400 & 5100    & 2500 & 3300 \\ \hline
charm   & 124 & 104     & 6100 & 6900    & 2800 & 4400 \\ \hline
top     & $<$0.1   & $<$0.1       & $<$0.1    & $<$0.1       & 0.3    & 0.5    \\ \hline
$W$     & $<$0.1   & $<$0.1       & 3.0    & 4.4       & 26   & 41    \\ \hline
TOTAL   & 663 & 377     & 19100 & 17300  &  6400 & 12900 \\ \hline
%\end{tabular}
%\begin{tabular}{|c|c|c|c|c|c|c|}
%\hline
& \multicolumn{2}{|c|} {5 GeV} & \multicolumn{2}{|c|} {8 GeV} & \multicolumn{2}{|c|} {40 GeV}   \\
%& Barrel (Hz) & Endcaps (Hz) & Barrel (Hz) & Endcaps (Hz) & Barrel (Hz) & Endcaps (Hz) \\
%\hline
$\pi/K$  & 162 & 81    & 2200   & 3800  & 470  & 1900  \\ \hline
beauty   & 54 & 53     & 2900  & 4000  & 1100 & 1300   \\ \hline
charm    & 76 & 73     & 3800  & 4700  & 1200 & 1400  \\ \hline
top      & $<$0.1 & $<$0.1       & $<$0.1     & $<$0.1     &  0.3   &  0.3 \\ \hline
$W$      & $<$0.1 & $<$0.1      & 4     & 4.5    & 23   & 33  \\ \hline
TOTAL    & 292 & 207  & 8900 & 12500 & 2800 & 4600  \\ \hline
\end{tabular}
\end{center}\vglue3mm
\caption{Single muon trigger rates at L1, for various low and high
\pt\ thresholds,
at  ${\cal{L}} = 10^{31}$ cm$^{-2}$ s$^{-1}$, ${\cal{L}} = 10^{33}$ cm$^{-2}$ s$^{-1}$ and ${\cal{L}} = 10^{34}$ cm$^{-2}$ s$^{-1}$. \label{L1:rates}}
\end{table}
\begin{table}[!htb]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multicolumn{7}{|c|}{L2 muon standalone trigger rates}\\\hline\hline
& \multicolumn{2}{|c|} {${\cal{L}} = 10^{31}$ cm$^{-2}$ s$^{-1}$} & \multicolumn{2}{|c|} {${\cal{L}} = 10^{33}$  cm$^{-2}$ s$^{-1}$}  & \multicolumn{2}{|c|} {${\cal{L}} = 10^{34}$ cm$^{-2}$ s$^{-1}$} \\
\hline
& Barrel (Hz) & Endcaps (Hz) & Barrel (Hz) & Endcaps (Hz) & Barrel (Hz) & Endcaps (Hz)  \\
\hline
& \multicolumn{2}{|c|} {4 GeV} & \multicolumn{2}{|c|} {6 GeV}  & \multicolumn{2}{|c|} {20 GeV} \\
%\hline
$\pi/K$  & 190 & 140 & 4300 & 3700 &410 & 1800 \\ \hline
beauty   &  50 &  67 & 3000 & 3900 &540 & 1500 \\ \hline
charm    &  70 &  94 & 4000 & 5200 &520 & 1700 \\ \hline
top     & $<$0.1  &  $<$0.1      &  $<$0.1   &  $<$0.1       & 0.2    & 0.4    \\ \hline
$W$     &  $<$0.1    &  $<$0.1       & 3    & 4       & 24   & 38    \\ \hline
%top     & $10^{-4}$   &  $10^{-4}$      & $10^{-2}$  & $10^{-2}$       & 0.2    & 0.4    \\ \hline
%$W$     & $10^{-2}$   &  $10^{-2}$      & 3    & 4       & 24   & 38    \\ \hline
TOTAL    & 310 & 301 &11300 &12800 &1494 & 5038\\ \hline
%\end{tabular}
%\begin{tabular}{|c|c|c|c|c|c|c|}
%\hline
& \multicolumn{2}{|c|} {5 GeV} & \multicolumn{2}{|c|} {8 GeV} & \multicolumn{2}{|c|} {40 GeV}   \\
%& Barrel (Hz) & Endcaps (Hz) & Barrel (Hz) & Endcaps (Hz) & Barrel (Hz) & Endcaps (Hz) \\
%\hline
$\pi/K$  &  82 &  120  &  840 & 1500   &  200 &  690   \\ \hline
beauty   &  37 &   59  & 1000 & 2200   &   87 &  280   \\ \hline
charm    &  49 &   81  & 1300 & 2900   &   83 &  290   \\ \hline
top     &  $<$0.1  & $<$0.1  & $<$0.1  & $<$0.1    &  0.1 &    0.2 \\ \hline
$W$     & $<$0.1  & $<$0.1   &   3 &    4 &  17 &   23 \\ \hline
TOTAL    & 168 &  260  & 3143 & 6604   &  387 & 1283   \\ \hline
\end{tabular}
\end{center}\vglue3mm
\caption{Single muon trigger rates at L2 muon standalone, for
various low and high \pt\ thresholds, at ${\cal{L}} = 10^{31}$
cm$^{-2}$ s$^{-1}$, $10^{33}$ cm$^{-2}$ s$^{-1}$ and $10^{34}$
cm$^{-2}$ s$^{-1}$. The large expected rate in particularly in the
endcap, caused by the relatively low rejection of low \pt\ muons,
can be reduced by improving the selection algorithm.
\label{L2mf:rates}}
\end{table}
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multicolumn{7}{|c|}{L2 muon combined trigger rates}\\\hline\hline
& \multicolumn{2}{|c|} {${\cal{L}} = 10^{31}$ cm$^{-2}$ s$^{-1}$} & \multicolumn{2}{|c|} {${\cal{L}} = 10^{33}$ cm$^{-2}$ s$^{-1}$}  & \multicolumn{2}{|c|} {${\cal{L}} = 10^{34}$ cm$^{-2}$ s$^{-1}$} \\
\hline
& Barrel (Hz) & Endcaps (Hz) & Barrel (Hz) & Endcaps (Hz) & Barrel (Hz) & Endcaps (Hz)  \\
\hline
& \multicolumn{2}{|c|} {4 GeV} & \multicolumn{2}{|c|} {6 GeV}  & \multicolumn{2}{|c|} {20 GeV} \\
%\hline
$\pi/K$  & 130     &  124 &3500 & 2600     &  68 &  890      \\ \hline
beauty   &   48    &   66 & 2700 & 3400    &  320 &  830    \\ \hline
charm    &    66   &   91 &  3800 & 4400   &   280 &  840   \\ \hline
top      & $<$0.1 & $<$0.1  &  $<$0.1  &$<$0.1  & 0.2  & 0.4 \\ \hline
$W$      & $<$0.1 & $<$0.1  & 3 & 4       & 22 & 35  \\ \hline
%top      &$10^{-4}$ & $10^{-4}$ &  $10^{-2}$ & $10^{-2}$  & 0.2  & 0.4 \\ \hline
%$W$      &$10^{-2}$ & $10^{-2}$ & 3 & 4       & 22 & 35  \\ \hline
TOTAL    &   244   &  281 & 10000 &11000   &   690 & 2590   \\ \hline
%\end{tabular}
%\begin{tabular}{|c|c|c|c|c|c|c|}
%\hline
& \multicolumn{2}{|c|} {5 GeV} & \multicolumn{2}{|c|} {8 GeV} & \multicolumn{2}{|c|} {40 GeV}   \\
%& Barrel (Hz) & Endcaps (Hz) & Barrel (Hz) & Endcaps (Hz) & Barrel (Hz) & Endcaps (Hz) \\
%\hline
$\pi/K$ &  44 &  55 & 400 &  530 &  6 &  310 \\ \hline
beauty  &  31 &  45 & 660 & 1100 & 31 &   92 \\ \hline
charm   &  41 &  61 & 780 & 1300 & 26 &   99 \\ \hline
%top     &$10^{-4}$  &  $10^{-4}$ & $10^{-2}$ & $10^{-2}$ & 0.1 & 0.1        \\ \hline
%$W$     &$10^{-2}$  &  $10^{-2}$    & 3 & 4        & 7 & 12   \\ \hline
top     & $<$0.1 &  $<$0.1  & $<$0.1 &  $<$0.1 & 0.1 & 0.1        \\ \hline
$W$     &  $<$0.1 &  $<$0.1  & 3 & 4        & 7 & 12   \\ \hline
TOTAL   & 116 & 161 &1840 & 2900 & 70 &  513 \\ \hline
\end{tabular}
\end{center}\vglue3mm
\caption{Single muon trigger rates at L2 muon combined, for  various
low and high \pt\ thresholds,
at ${\cal{L}} = 10^{31}$ cm$^{-2}$ s$^{-1}$, $10^{33}$ cm$^{-2}$ s$^{-1}$, and $10^{34}$ cm$^{-2}$ s$^{-1}$. \label{L2mc:rates}}
\end{table}
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multicolumn{7}{|c|}{Event Filter muon trigger rates}\\\hline\hline
& \multicolumn{2}{|c|} {${\cal{L}} = 10^{31}$ cm$^{-2}$ s$^{-1}$} & \multicolumn{2}{|c|} {${\cal{L}} = 10^{33}$ cm$^{-2}$ s$^{-1}$}  & \multicolumn{2}{|c|} {${\cal{L}} = 10^{34}$ cm$^{-2}$ s$^{-1}$} \\
\hline
& Barrel (Hz) & Endcaps (Hz) & Barrel (Hz) & Endcaps (Hz) & Barrel (Hz) & Endcaps (Hz)  \\
\hline
& \multicolumn{2}{|c|} {4 GeV} & \multicolumn{2}{|c|} {6 GeV}  & \multicolumn{2}{|c|} {20 GeV} \\
%\hline
$\pi/K$  & 125 & 119 & 1890 & 1230         & 46 & 40  \\ \hline
beauty   & 44 & 56   & 1870 & 2190         & 260 & 380\\ \hline
charm    & 60 & 76   & 2390 & 2780         & 220 & 330\\ \hline
top      &$<$~0.1 & $<$~0.1  &$<$~0.1 &$<$~0.1  &0.2 & 0.3\\ \hline
$W$      &$<$~0.1 &$<$~0.1      & 2.9 & 3.9    & 21 & 31  \\ \hline
TOTAL    & 229 & 251 & 6150 & 6200         & 550 & 780\\ \hline
%\end{tabular}
%\begin{tabular}{|c|c|c|c|c|c|c|}
%\hline
& \multicolumn{2}{|c|} {5 GeV} & \multicolumn{2}{|c|} {8 GeV} & \multicolumn{2}{|c|} {40 GeV}   \\
%& Barrel (Hz) & Endcaps (Hz) & Barrel (Hz) & Endcaps (Hz) & Barrel (Hz) & Endcaps (Hz) \\
%\hline
$\pi/K$ & 36 & 25           & 290 & 260    & 0.14 & 0.2   \\ \hline
beauty  & 27 & 33           & 550 & 800    & 10.5 & 16.3 \\ \hline
charm   & 36 & 43           & 640 & 930    & 7.1 & 11.1  \\ \hline
top     &$<$~0.1 &$<$~0.1   &$<$~0.1 &$<$~0.1  & $<$~0.1 & $<$~0.1        \\ \hline
$W$     &$<$~0.1   &$<$~0.1     & 2.8 & 3.8    & 3.9 & 6.1   \\ \hline
TOTAL   & 99 & 101          & 1480 & 1990  & 21.7 & 33.7 \\ \hline
\end{tabular}
\end{center}\vglue3mm
\caption{Single muon trigger rates at EF muon combined, for various
low and high \pt\ thresholds,
at ${\cal{L}} = 10^{31}$ cm$^{-2}$ s$^{-1}$,  $10^{33}$ cm$^{-2}$ s$^{-1}$, and  $10^{34}$ cm$^{-2}$ s$^{-1}$. \label{EF:rates}}
\end{table}

The trigger rates for single muon event originating from all the
physical processes expected in ATLAS were obtained using the EF MuiD
combined algorithm (the events must of course also pass the L1 and
L2 muon algorithms). Various luminosity scenarios expected during
LHC operation (from ${\cal{L}} = 10^{31}$ cm$^{-2}$ s$^{-1}$ to
$10^{34}$ cm$^{-2}$ s$^{-1}$) were considered. Trigger rates were
typically computed by convolving, over a given \pt\ range,  the
estimated efficiencies with the cross-sections of processes
representing the main muon sources at LHC.
%({\it convolution method})
For the  i-th process with cross-section $\sigma_i$, the rate is

\begin{equation}
R_i = {\cal{L}} \int  \frac{d\sigma_i}{d p_T} \epsilon(p_T) dp_T
%\limits^{p_{T}^{inf}}_{p_{T}^{cutoff}}
\label{eq:rates}
\end{equation}

\noindent where  {$\cal{L}$} is the instantaneous luminosity and
$\epsilon(p_T)$ is the muon trigger efficiency for a given \pt\
value. In order to take into account the  $\eta$ dependence,
separate estimates for different $\eta$ regions have been
considered. An 0.5 GeV step has been applied in the numerical
integration. The inclusive muon cross-sections at the  LHC for
$b\rightarrow\mu$ and $c\rightarrow\mu$ decays have been
parameterized by using PYTHIA 6.403 \cite{pythia6} which produces
conservative estimates
since it predicts
cross-sections  about 2 to 3 times higher than previous descriptions \cite{ATLASHLTTDR}.
Top quark and W/Z decays were simulated using
PYTHIA 5.7 \cite{pythia5}.
Rates of muon in-flight decays from  $\pi$/$K$ mesons have been computed using the
DPMJET Monte Carlo program \cite{dpmjet}.

To verify the results obtained with this method and to understand
the systematics, an alternative approach, relying on  event
counting, has been applied to the minimum bias events ({\it counting
method}). The convolution and counting methods give EF final rates
which are in good agreement, within statistical errors due to the
limited size of the minimum bias sample, starting from \pt\
threshold of 6 GeV. The values obtained with the counting method for
lower \pt\ thresholds (4 and 5 GeV) are a factor of two to four less
for muons from $\pi$/$K$ decays with respect to the convolution
(provided by DPMJET) of Eq. \ref{eq:rates}.

The rates obtained for some low and high \pt\ thresholds in the
barrel and in the endcaps after L1,  L2 muFast, L2 muComb and EF
selection are shown in Tables \ref{L1:rates}, \ref{L2mf:rates},
\ref{L2mc:rates} and \ref{EF:rates}.
%The results reported here  supersede those previously
%obtained  \cite{ATLASHLTTDR} as they are based on an updated parametrization of cross
%section of muons coming from  b and c quark decays.

In Fig.~\ref{EF:plot_rate} the total (barrel+endcaps) EF rates at L
= $10^{31}$ cm$^{-2}$ s$^{-1}$  are shown as a function of the \pt\
threshold. In this figure, to keep uniformity among the rate
results, mostly provided by PYTHIA 6.403, it has been chosen to
report for the 4 and 5 GeV thresholds the EF rates obtained with the
counting procedure.
\begin{figure}[!t]\begin{center}
\includegraphics[width=0.65\figwidth]{fig/EFMuons.eps}
%\includegraphics[width=0.55\figwidth]{fig/EF_muonrates.last.epsi}
\caption{Expected EF rates at ${\cal{L}} = 10^{31}$ cm$^{-2}$
s$^{-1}$ for single muon processes as a function
of muon $\pt$ threshold integrated over $|\eta| < 2.4$.
\label{EF:plot_rate}}
\end{center}\end{figure}

%For the {\em convolution} method, it is crucial to correctly predict
%the differential cross-sections of muons and therefore the production
%cross-section for $\pi/K$ and beauty and charm hadrons. The comparison
%for $\pi/K$ has already been shown in Fig.\ref{fig:min_bias_pi} and
%Fig.\ref{fig:min_bias_K} showing a good agreement. The comparison
%between the differential
%cross-section from PYTHIA and FONLL\cite{FONLL} for beauty and charm hadrons are
%shown respectively in Fig.\ref{fig:min_bias_fonll_charm} and
%Fig.\ref{fig:min_bias_fonll_beauty}.  {\bf Quando la
%farm \e di nuovo su devo paragonare queste sezioni d'urto a quelle da
%Pythia bbmu6X}

%\begin{figure}[htbp]
%  \begin{center}
%    \includegraphics[width=0.7\columnwidth]{fig/fonll_charm_bare.eps}
%  \end{center}
%  \caption{Differential cross-section of charm bare'' quark from
%FONLL and PYTHIA. $|y|<2.0$, CTEQ6M, $m_c= 1.5$ GeV/$c^2$ for charm,
%$\mu_{R} = \mu_F = \mu_0 = \sqrt{m^2 + P_T^2}$, theoretical
%uncertainties include scale uncertainties($\mu_0/2 < \mu_R$ and $\mu_F < 2\mu_0$ with $1/2 < \mu_R/\mu_F < 2$.
%and mass uncertainties ($1.3< m_c<1.7$) summed in quadrature.
%  \label{fig:min_bias_fonll_charm}}
%\end{figure}

%\begin{figure}[htbp]
%  \begin{center}
%    \includegraphics[width=0.7\columnwidth]{fig/fonll_beauty_bare.eps}
%  \end{center}
%  \caption{Differential cross-section of charm bare'' quark from
%FONLL and PYTHIA. $|y|<2.0$, CTEQ6M, $m_b= 4.75$ GeV/$c^2$ for charm,
%$\mu_{R} = \mu_F = \mu_0 = \sqrt{m^2 + P_T^2}$, theoretical
%uncertainties include scale uncertainties($\mu_0/2 < \mu_R$ and $\mu_F < 2\mu_0$ with $1/2 < \mu_R/\mu_F < 2$.
%and mass uncertainties ($4.5< m_b<5.0$) summed in quadrature.
%  \label{fig:min_bias_fonll_beauty}}
%\end{figure}

\subsection{Fake dimuon trigger rate}

A single muon can be detected in multiple muon trigger sectors,
causing such events to erroneously satisfy dimuon triggers. Such
fake triggers can be suppressed by the overlap handling implemented
in the MuCTPI . For the single muon samples used in the analysis,
the fake dimuon trigger probability is defined as

\begin{equation}
P_{\rm fake} = \frac{\rm Number~of~events~with~more~than~one~muon~triggered}
{\rm Number~of~events~with~a~triggered~muon}
\end{equation}

%The fake dimuon trigger probabilities have been calculated for both
%the nominal and 900 GeV'' threshold configurations.
Four sources of fake double-counts have been considered :
%when a single muon is detected
%    by two overlapping RPC sectors (Barrel-Barrel double counts, BB),
%    by an overlapping RPC-TGC sector pair (Barrel-Endcap double counts, BE),
%   by two overlapping Endcap'' TGC sectors (Endcap-Endcap double counts, EE) and
%by two overlapping Forward'' TGC sectors (Forward-Forward double counts, FF)
\begin{itemize}

\item Barrel-Barrel double counts (BB): When a single muon is detected
by two overlapping RPC sectors.

\item Barrel-Endcap double counts (BE): When a single muon is detected
by an overlapping RPC-TGC sector pair.

\item Endcap-Endcap double counts (EE): When a single muon is detected
by two overlapping Endcap'' TGC sectors.

\item Forward-Forward double counts (FF): When a single muon is detected
by two overlapping Forward'' TGC sectors.

\end{itemize}

The probabilities that a single muon would cause any of these fake
double counts have been calculated separately.  The effect of the
MuCTPI  overlap handling can be seen in Fig.~\ref{fig:MuCTPI:be_prob}, where the left plot
shows the BE fake dimuon
trigger probabilities for the 6 $\pt$ thresholds in the 2  to 50~GeV
range without using the overlap handling, while the right plot shows
the probabilities after applying the  MuCTPI  overlap handling.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\columnwidth]{fig/muctpi_probability.eps}
\end{center}
\icaption{Barrel-Endcap fake dimuon trigger probabilities without (a) and
with (b) using the overlap handling of the MuCTPI.
\label{fig:MuCTPI:be_prob}}
\end{figure}

The  probabilities for 6 and 20~GeV single muons to produce a fake
dimuon trigger if they caused a single-muon trigger, for all
available L1 muon thresholds, can be seen in Table
\ref{tab:MuCTPI:prob1}.

\begin{table}[!ht]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
%\multicolumn{6}{|c|}{Fake double muon probability} \\

%\hline
Trigger item & \pt\~[GeV] & BB prob. [\%] & BE prob. [\%] & EE prob. [\%] & FF prob. [\%] \\
\hline
\hline
2mu4 & 6.0 &  1.56 $\pm$ 0.07 &  1.39 $\pm$ 0.08 &  1.00 $\pm$ 0.07 &  0.81 $\pm$ 0.06 \\
& 20.0 &  1.43 $\pm$ 0.06 &  0.13 $\pm$ 0.02 &  0.49 $\pm$ 0.05 &  0.55 $\pm$ 0.05 \\
\hline
\hline
2mu5 & 6.0 &  1.14 $\pm$ 0.06 &  1.17 $\pm$ 0.07 &  0.40 $\pm$ 0.05 &  0.56 $\pm$ 0.06 \\
& 20.0 &  1.43 $\pm$ 0.06 &  0.13 $\pm$ 0.02 &  0.49 $\pm$ 0.05 &  0.55 $\pm$ 0.05 \\
\hline
\hline
2mu6 &6.0 &  1.11 $\pm$ 0.05 &  0.97 $\pm$ 0.06 &  0.39 $\pm$ 0.04 &  0.55 $\pm$ 0.05 \\
& 20.0 &  1.43 $\pm$ 0.06 &  0.13 $\pm$ 0.02 &  0.49 $\pm$ 0.05 &  0.55 $\pm$ 0.05 \\
\hline
\hline
2mu8 &6.0 &  0.87 $\pm$ 0.05 &  0.38 $\pm$ 0.06 &  0.31 $\pm$ 0.05 &  0.58 $\pm$ 0.07 \\
& 20.0 &  1.33 $\pm$ 0.06 &  0.10 $\pm$ 0.02 &  0.42 $\pm$ 0.05 &  0.45 $\pm$ 0.05 \\
\hline
\hline
2mu10 & 6.0 &  0.68 $\pm$ 0.05 &  0.12 $\pm$ 0.08 &  0.21 $\pm$ 0.09 &  0.54 $\pm$ 0.13 \\
&20.0 &  1.26 $\pm$ 0.06 &  0.10 $\pm$ 0.02 &  0.36 $\pm$ 0.04 &  0.36 $\pm$ 0.04 \\
\hline
\hline
2mu11 & 6.0 &  0.43 $\pm$ 0.21 &  0.00 $\pm$ 0.00 &  0.32 $\pm$ 0.15 &  0.42 $\pm$ 0.16 \\
& 20.0 &  0.86 $\pm$ 0.05 &  0.00 $\pm$ 0.00 &  0.33 $\pm$ 0.04 &  0.32 $\pm$ 0.04 \\
\hline
\hline
2mu20 &6.0 &  0.28 $\pm$ 0.28 &  0.00 $\pm$ 0.00 &  0.48 $\pm$ 0.34 &  0.00 $\pm$ 0.00 \\
& 20.0 &  0.75 $\pm$ 0.05 &  0.00 $\pm$ 0.00 &  0.24 $\pm$ 0.03 &  0.18 $\pm$ 0.03 \\
\hline
\hline
2mu40 &6.0 &  0.42 $\pm$ 0.42 &  0.00 $\pm$ 0.00 &  0.00 $\pm$ 0.00 &  0.00 $\pm$ 0.00 \\
& 20.0 &  0.49 $\pm$ 0.04 &  0.00 $\pm$ 0.00 &  0.08 $\pm$ 0.03 &  0.08 $\pm$ 0.03 \\
\hline

\end{tabular}
\end{center}
\caption{Probabilities that single muons with transverse
momenta 6 and 20 GeV which caused a single muon trigger,
to also passes a fake dimuon signature at the same
threshold.
\label{tab:MuCTPI:prob1}}
\end{table}
%To calculate the fake dimuon trigger rates with a
%modified version of the code used to calculate single-muon trigger
%rates, these probabilities have to be calculated for the 4 different
%$\eta$ regions of the Muon Spectrometer separately.
The fake probabilities can be used to calculate the single-muon
trigger rates according to

\begin{equation}
R_{\rm fake} = {\cal{L}} \int \limits^{p_{T}^{inf}}_{p_{T}^{cutoff}} \sigma_p(p_T)
\epsilon(p_T) P_{\rm fake}(p_T) dp_T
\end{equation}

\noindent where ${\cal{L}}$ is the instantaneous luminosity of the
accelerator, $\sigma_p$ is the inclusive muon production
cross-section at LHC and $\epsilon$ is the L1 trigger efficiency.
%Using this formula on the
%trigger efficiencies provided by the RPC and TGC groups and the fake
%dimuon trigger probabilities acquired in this analysis, one gets the
The fake dimuon trigger rates are presented in Table
\ref{tab:MuCTPI:rates}.
\begin{table}[ht]
\renewcommand{\arraystretch}{1.2}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}

\hline
Trigger item & BB rate [Hz] & BE rate [Hz] & EE rate [Hz] & FF rate [Hz] & Total fake rate [Hz] \\
\hline
2mu4 & 1846.6 $\pm$ 119.2 & 271.6 $\pm$ 14.1 & 136.2 $\pm$ 24.5 & 69.2 $\pm$ 12.3 & 2323.7 $\pm$ 123.1 \\
2mu5 & 243.9 $\pm$ 13.0 & 203.1 $\pm$ 10.6 & 35.3 $\pm$ 10.5 & 33.3 $\pm$ 6.5 & 515.5 $\pm$ 20.8 \\
2mu6 & 193.9 $\pm$ 12.4 & 82.6 $\pm$ 7.1 & 24.6 $\pm$ 6.0 & 24.7 $\pm$ 4.7 & 325.7 $\pm$ 16.2 \\
2mu8 & 114.1 $\pm$ 9.8 & 16.1 $\pm$ 2.1 & 9.7 $\pm$ 3.0 & 12.4 $\pm$ 3.3 & 152.3 $\pm$ 11.0 \\
2mu10 & 79.2 $\pm$ 8.0 & 4.9 $\pm$ 1.2 & 4.8 $\pm$ 2.2 & 5.5 $\pm$ 2.0 & 94.4 $\pm$ 8.6 \\
2mu11 & 11.7 $\pm$ 1.8 & 0.1 $\pm$ 0.1 & 3.9 $\pm$ 2.0 & 4.5 $\pm$ 1.8 & 20.1 $\pm$ 3.2 \\
2mu20 & 2.4 $\pm$ 0.4 & 0.1 $\pm$ 0.0 & 2.4 $\pm$ 1.9 & 0.7 $\pm$ 0.5 & 5.5 $\pm$ 2.0 \\
2mu40 & 0.8 $\pm$ 0.1 & 0.0 $\pm$ 0.0 & 1.7 $\pm$ 1.7 & 0.1 $\pm$ 0.1 & 2.6 $\pm$ 1.7 \\
\hline

\end{tabular}
\end{center}
\caption{Rates of various fake dimuon triggers at
${\cal{L}} = 10^{33}$ cm$^{-2}$ s$^{-1}$.
\label{tab:MuCTPI:rates}}
\end{table}

\section{Rejection of muons from $\pi$/$K$ decays }
\label{EF_pi_K}

Despite the large theoretical uncertainties on the rates of the muon
production processes at low transverse momentum, it is clear that in
flight decays of pions and kaons are a significant  source of single
muons and, therefore, a strategy must be developed to reject these
events in the trigger.
%trigger rate at low transverse momentum and, therefore, deserve the
%study of a dedicated rejection strategy.
Rejection of muons from $\pi$ and $K$ decays at the EF is described below. A study
describing rejection at L2 can be found in  \cite{BphyKPI}.
%In the following only the analysis for Event Filter is reported; for L2
%see \cite{BphyKPI}.

\subsection{Data samples and their validation}

Minimum bias samples would be the most suitable for studies involving pion and kaon decays.
%The most suitable simulations to perform such studies would be minimum
%bias samples;
However, the probability that pions or kaons produced in
low or moderate $\pt$ QCD scattering would decay before
interacting hadronically in the calorimeters is low, between 0.1\% and
1\% depending on the meson $\pt$.
%Therefore, a dedicated tool has
%been developed
In order to enhance the
number of charged pion and kaon decays in the sample, the simulation
of events without any charged meson with $\pt$ above a given
threshold is aborted and one $\pi^{\pm}$ or $K^{\pm}$ is forced to
decay in the Inner Detector cavity.

The samples produced are:
\begin{itemize}
\item $10^6$ single pions with $\pt >2.5~GeV$ and kinematics
($\pt\times\eta$) generated according to a double differential
cross-section of primary pions in minimum bias events
%observed in
%   previous simulations, forced to decay;
\item $10^5$ minimum bias events, where one charged $\pi$ or $\rm K$
with $\pt >2~GeV$ per event is forced to decay;
\end{itemize}
%A special care must be payed when using the above samples
In order to estimate
cross-sections or trigger rates, the abundance of forced decays must be
re-weighted on an event by event basis according to meson decay probability.
In addition to the above samples, standard minimum bias events have
been used as a reference to cross check the results obtained
from these dedicated productions.

% \begin{table}
% \begin{tabular}{c||c|c||c|c||c|c||c|c}
%   & & & & & \multicolumn{2}{c||}{Std min bias} & \multicolumn{2}{c||}{Forced min bias}\\
%   $\rm \sigma (\mu b)$ & \multicolumn{2}{c||}{PYTHIA 5.7} &
%   \multicolumn{2}{c||}{DPMJET-II} & \multicolumn{2}{c||}{PYTHIA 6.4} &
%   \multicolumn{2}{c}{PYTHIA 6.4} \\
% & $\pi^\pm$ & $K$ &  $\pi^\pm$ & $K$ &  $\pi^\pm$ &  $K^\pm$ &  $\pi^\pm$ &  $K^\pm$ \\ \hline
% %%$p_T >4 GeV, |\eta|<2.7$ &  6.4  & 4.3  & 9.5  & 8.6  & & & & \\
% $\rm {p_T}_\mu >4~ GeV$ &  5.7   & 4.1   & 9.1   & 8.2   &  5.4$\pm$ 1.2  &
%   8.9 $\pm$ 1.5  & 5.7 $\pm$ 1.0 & 5.4 $\pm$ 1.1 \\ \hline
% %%$p_T >6 GeV, |\eta|<2.7$ &  0.83  & 0.64  & 1.21  & 1.07  & & & & \\
% $\rm {p_T}_\mu >6~ GeV$ &  0.79  & 0.61  & 1.16  & 1.03  &  0.8$\pm$ 0.5  &
% 1.3 $\pm$ 0.6  & 1.2 $\pm$ 0.6 & 1.9 $\pm$ 0.6 \\
% \end{tabular}
%\begin{table}
%\begin{tabular}{c||c|c||c|c||c|c||c|c||}
% & \multicolumn{2}{c||}{} & \multicolumn{2}{c||}{} & \multicolumn{2}{c||}{Std min bias} & \multicolumn{2}{c||}{Forced min bias} \\
%  $\rm \sigma (\mu b)$ & \multicolumn{2}{c||}{PYTHIA 5.7} &
%  \multicolumn{2}{c||}{DPMJET-II} & \multicolumn{2}{c||}{PYTHIA 6.4} &
%  \multicolumn{2}{c||}{PYTHIA 6.4} \\
%& $\pi^\pm$ & $K$ &  $\pi^\pm$ & $K$ &  $\pi^\pm$ &  $K^\pm$ &  $\pi^\pm$ &  $K^\pm$ \\ \hline
%%%$p_T >4 GeV, |\eta|<2.7$ &  6.4  & 4.3  & 9.5  & 8.6  & & & & \\
%$\rm {p_T}_\mu >4~ GeV$ &  5.7   & 4.1   & 9.1   & 8.2   &  5.4$\pm$ 1.2  &
%  8.9 $\pm$ 1.5  & 5.7 $\pm$ 1.0 & 5.4 $\pm$ 1.1 \\ \hline
%%%$p_T >6 GeV, |\eta|<2.7$ &  0.83  & 0.64  & 1.21  & 1.07  & & & & \\
%$\rm {p_T}_\mu >6~ GeV$ &  0.79  & 0.61  & 1.16  & 1.03  &  0.8$\pm$ 0.5  &
%1.3 $\pm$ 0.6  & 1.2 $\pm$ 0.6 & 1.9 $\pm$ 0.6 \\
%\end{tabular}
%\caption{A comparison of predictions for muon production cross
%  sections from decays in flight of light mesons within kinematic
%  regions of interest for the muon trigger (a cut to $|\eta_\mu|<2.4$
%  is applied). A normalization of 80~mb is assumed, in all cases, for
%  the total inelastic minimum bias cross-section.  The first and
%  second columns report the results obtained by integrating the
%  parametrization of the double differential cross-section
%  historically used to estimate muon trigger rates. The cross-section
%  observed on limited statistics of standard minimum bias simulation,
%  by event counting, is reported in column three.  Finally, the
%  predictions from the sample of minimum bias events with enhanced
%  $\pi/K$ decays are also shown.\label{tabCrossSectPiK}}
%\end{table}

The muon $\pt$  spectra  observed in minimum bias
events and single pions, forced to decay, were found to be
consistent, after appropriate re-weighting, with each other and in
agreement with previous predictions and unforced minimum bias events.
%In order to make the comparison
%between the samples more explicit, Table~\ref{tabCrossSectPiK}
%summarizes the cross-sections predicted for $\rm p_T>4$ and $\rm %6~GeV$ in the pseudorapidity acceptance of the muon L1 trigger. In
%spite of the large statistical uncertainties and of possible
%systematic biases induced by the treatment of forced decays, the
%discrepancies observed are below the large statistical uncertainties
%on the low $\rm p_T$ physics underlying this kind of processes and,
%therefore, they can be considered of minor importance for the purposes
%of trigger studies.

%\subsection{Rejection strategy at Level 2}
%Reconstruction of muons coming from in flight decays of $\pi/K$ has a
%different efficiency with respect to prompt muons. In
%Figure \ref{fig:level2_pik_rejection-eff} we show as an example, the {\em mu6}
%efficiency as a function of
%transverse momentum for prompt muons, muons from $\pi$ and $K$
%mesons. The efficiency curve for muons from $\pi/K$ is sensibly lower
%than the efficiency curve for prompt muons.
%
%Some additional selection have been studied trying to  increase the
%rejection of $\pi$ and $K$ while keeping a large efficiency for prompt
%muons. In Fig.~\ref{fig:level2_pik_rejection-rej} we show the efficiency as a
%function of the rejection power ($1-\mathrm{Eff(Bkg)}$) obtained
%varying the matching cut between the
%momentum measured in the MS and the one measured in the
%ID\footnote{The quantity on which a further selection is applied is
%$\chi^2= %\frac{(P_{T_{MS}}-P_{T_{ID}})^2} %{\sigma^2_{P_T{_{MS}}}+\sigma^2_{P_{T_{ID}}}}$.}. The
%efficiencies have been obtained on a sample of prompt muons with
%$p_{T}=11$\, GeV and from the Pythia {\tt bbmu6X} sample while the rejections have been obtained on the {\em
%Minimum Bias + Pion Decayer} sample. Since the rejection  doesn't seem
%to improve this further selection was
%not implemented in the following analysis. However, we expect some
%improvements with more recent ATLAS trigger simulation versions.

%\begin{figure}[htbp]
%\centering
%\subfigure[Efficiency {\em vs} transverse momentum of {\em mu6}
%selection: muFast , muComb on prompt muons, muComb on muons from
%in-flight decays ($pi\/K$)]{\label{fig:level2_pik_rejection-eff}
%\includegraphics[width=0.45\textwidth]{fig/eff_mucomb_pik.eps}}
%\subfigure[Efficiency {\em  vs} rejection ($1-\mathrm{Eff_{Bkg}}$) varying the
%   matching cut
%between the track reconstructed in the MS and the one in the ID]{\label{fig:level2_pik_rejection-rej}
%\includegraphics[width=0.45\textwidth]{fig/eff_vs_rej.eps}}
%\caption{L2 $pi$/K rejection}
%\label{fig:level2_pik_rejection}
%\end{figure}

\subsection{Rejection strategy at the event filter}
The fraction of in flight decay muons retained at the EF, normalised
to the L2 efficiency, for the mu6 trigger item, has been measured as
a function of the muon $\pt$.
% by processing the single pion
%sample with the standard L1 and HLT emulation chain.
There is  a very poor rejection capability ($\lesssim 90\%$) for muons coming
from pion decays, which demonstrates that the standard muon
identification procedures are not very sensitive, as expected, to the
small kink between the pion and muon tracks. The kinematics of charged
kaon two-body decays, which are the dominating kaon contribution to
the muon rate, is much more favorable toward rejection due to
the larger average value of the angle between the kaon and the muon
tracks. In order to improve the rejection capability, additional
measured parameters providing some discriminating power between
background and primary muons have been identified:
\begin{itemize}
\item the impact parameter, $\rm d_0$, of the track reconstructed in the inner
tracker;
%the width of the distribution of such parameter depends on the
%resolution of the ID reconstruction; low quality reconstructed
%tracks,with hits produced before and after the decay kink, exhibit spoiled
%resolution on the impact parameter. Moreover, in case only the hits associated to the
%muon track are actually used in the fit, the impact parameter would be an
%indirect measurement of the decay kink;
\item
% as a consequence of the kink some hits in the ID, most probably those at
%the entrance of the inner tracker, might constribute with high
%residuals to the track and, therefore, they might be discarded by the fitting
%procedure; for this reason
the number of hits associated to the Inner Detector track in the
Pixel Detector ($\rm N_{hits}(Pixel)$), in the pixel B-layer ($\rm N_{hits}(B_{layer})$)
and in the Silicon Tracker ($\rm N_{hits}(SCT)$);
% have been studied in single
%muons samples and in muons from pion decays;
\item the ratio $\rm {p_T}_{ID}/{p_T}_{MS}$ between the transverse $\pt$ in the Inner
Detector and in the Muon Spectrometer, after back-extrapolation to the
interaction point and correction for the measured energy loss in the
calorimeters;
%, is ideally simmetrically distributed around one; in case of fake
%muons from pions, in addition to a degradation of the resolution in the inner
%tracker, the kinematic mismatch of transverse momentum between the decaying meson and
%the muon can lead to a tail at high values; due to the steeply falling $\rm %p_T$ spectrum of charged particles in minimum bias events, this behaviour is enhanced at high
%transverse momentum;
\item the $\rm \chi^2_{matching}$ of the matching between the track parameters as reconstructed in the Muon
Spectrometer and in the Inner Detector.
% is clearly a
%valuable discriminating variable, in case the ID track is mostly built out of
%hits produced by the decaying meson.
\end{itemize}
The discrimination power of each variable has been studied by
measuring the fraction of accepted events as a function of the cut
applied for both isolated muons and fake muons above a given
$\pt$ threshold.
The results, shown in Fig.~\ref{figDiscriminatingCuts}, are based on the simulations of single
muons and single pions with forced decays. For each variable, the
fraction of events retained after the cut is normalised to the
number of events passing the EF reconstruction before the
application of any hypothesis algorithms.
%\footnote{In addition, only
%events with a single muon reconstructed at the EF and a single track
%in the Inner Detector are considered in the reference sample. These simplifying
%conditions, which leave  the single muon
%sample almost unaffected, are considered to affect the result of the study in the
%direction of a more conservative estimate of the rejection power of
% muons from $\pi$.}.
\begin{figure}[th!]
\begin{center}
\includegraphics[width=0.9\columnwidth]{fig/allCutsEfficienciesBW_new.eps}
\end{center}
\caption{Efficiency for prompt muons and muons from pion decays as a function
of the cut on some discriminating variables.}
\label{figDiscriminatingCuts}
\end{figure}
From the analysis of the exclusive rejection power of the individual
variables, the set of cuts listed below have been defined. These cuts  try to minimize
the efficiency loss for prompt muons while reducing the background:
\begin{itemize}
\item $\rm |d_0| < 0.15~mm$, $\rm N_{hits}(B_{layer})\ge 1$, $\rm N_{hits}(Pixel)\ge 3$, $\rm N_{hits}(SCT)\ge 6$,
\item $\rm {p_T}_{ID}/{p_T}_{MS} < 1.25$, $\rm \chi^2_{matching} \le 26$.
\end{itemize}
In particular, these values have been chosen by considering
efficiency and background rejection at $\pt = 4~GeV$. It is assumed
that cuts will be optimized for each muon item in the trigger menu.

From the application of these cuts on the reference sample of events
accepted at the EF, the efficiency for prompt muons and in flight decay muons
shown in Fig.~\ref{figEffCutsMuPi} have been obtained.
\begin{figure}[!h]
\begin{center}

\includegraphics[width=0.54\figwidth]{fig/muEffFinalCuts.eps}
\includegraphics[width=0.54\figwidth]{fig/piEffFinalCuts.eps}
\includegraphics[width=0.54\figwidth]{fig/minBiasEffFinalCuts.eps}
\includegraphics[width=0.54\figwidth]{fig/minBiasPiKEffFinalCuts.eps}
\end{center}
\caption{ EF efficiency as a function of $\pt$ for different rejection cuts for
prompt muons (a), muons from single pion decays (b), minimum bias events (c and d).
In (c)  $\pi/K$ contribution has been separated. Each efficiency
curve shows
the data reduction obtained by the addition of the
corresponding cut to the overall selection procedure. The
specific values of the cuts are discussed in the
text.\label{figEffCutsMuPi}}
\end{figure}
A loss of efficiency between 25\% at the 4~GeV threshold and 10\% at
20~GeV correspond to a reduction in background of 65\% and 75\%,
respectively.  The rejection achieved for kaon decays is slightly
better than that achieved for $\pi$ decays, as expected from the
different decay kinematics. These results are derived from nominal
detector performance and algorithm resolutions. However, they
demonstrate that cuts can be adjusted to obtain reasonable trigger
rate at the very low $\pt$ threshold of  4 GeV which is reached
mostly by reducing  uninteresting events at the cost of some
efficiency loss for prompt muons. An optimization of the cuts, with
specific tuning for each trigger element, will eventually further
improve the signal to background ratio.

\section{Muon isolation}
\label{sec:muiso_perf}
\subsection{Optimization procedure}

The L2 isolation algorithm is seeded by either muFast or muComb. The
algorithm decodes  LAr and Tile Calorimeter quantities (i.e.
transverse energy deposit or sums of calorimetric cells above a
predefined energy threshold) in cones centered around the muon
direction. The geometrical definition of these cones is given by the
condition $\Delta R < \Delta R_{MAX}$, where $\Delta R = \sqrt(\Delta\eta^2+\Delta\phi^2)$, and $\Delta\eta$, $\Delta\phi$
are the distances in pseudorapidity and azimuthal angle between the
calorimetric cell and the cone axis. Because the muon itself
contributes to the energy deposit inside the cone, to improve the
discriminating power of the isolation algorithm, two different
concentric cones are defined: an internal cone chosen to contain the
energy deposit released by the muon itself, and an external one,
supposed to include contributions only from detector noise, pile-up
and jet particles if present. The optimization of the muon isolation
algorithm consists of determining the optimal size of the inner and
outer cone radius, the values of the cell energy thresholds, used to
compute the transverse energy and number of cells sums, and the
isolation requirements.
%to be applied in the hypothesis testing algorithm $MuIsoHypo$.

Table~\ref{tab:muiso1} summarizes the samples that have been used to
optimize the algorithms and to measure their performance. Half of
the events in the samples number 1 and 2 in the Table have been used
as signal and background, respectively, in the optimization of the
algorithm parameters, the remaining events for sample 1 and 2 and
the other samples listed in the Table have instead been used to
estimate the algorithm performances.

%It is important to observe here that at present
Only the parameters relative at the muon trigger in the barrel region ($|\eta|<1.05$)
have been studied.
% optimization of the algorithm in the forward region is ongoing and will be the subject of a future note.
%Both samples use full simulation of the ATLAS detectors with official software release and parameters for the CSC note production.
Simulation of the electronic readout noise for both LAr and Tile Calorimeters has been also included.

\begin{table}
\begin{center}
\begin{tabular}{|c|l|l|c|}\hline
& Process  & Generator & Number of events \\\hline
1& $Z\rightarrow\mu^+\mu^-$         & {\tt Pythia} &  $1~10^4$ \\\hline
2 &$b\bar{b}\rightarrow\mu(15)X$    & {\tt Pythia} &  $1.5~10^4$ \\\hline
3 &$b\bar{b}\rightarrow\mu(6)X$     & {\tt Pythia} &  $1~10^4$ \\\hline
4& $q\bar{q}\rightarrow\mu X$       & {\tt Pythia} &  $2.1~10^4$ \\\hline
5 & Single-$\mu$($\pt$=100~GeV) & {\tt Single-Mu gun} & $2~10^5$ \\\hline
6 &Single-$\mu$($\pt$=38~GeV)  & {\tt Single-Mu gun} & $2~10^5$ \\\hline
7 &Single-$\mu$($\pt$=19~GeV)  & {\tt Single-Mu gun} & $2~10^5$  \\\hline
%Process  & Dataset ID & Number of events \\\hline
%$Z\rightarrow\mu^+\mu^-$         & {\tt Pythia-Zmumu misal1\_csc11 005145 v12003101} &  $1~10^4$ \\\hline
%$b\bar{b}\rightarrow\mu(15)X$    & {\tt Pythia-BBar  misal1\_mc12  005701 v12000604} &  $1.5~10^4$ \\\hline
%$b\bar{b}\rightarrow\mu(6)X$     & {\tt Pythia-BBar  misal1\_mc12  017500 v12003106} &  $1~10^4$ \\\hline
%$q\bar{q}\rightarrow\mu X$       & {\tt Pythia-J4mu  misal1\_mc12  008070 v12000605} &  $2.1~10^4$ \\\hline
%$mu100$ & {\tt Single-Mu misal1\ mc12  007217 v12003107} & $2~10^5$ \\\hline
%$mu38$  & {\tt Single-Mu misal1\ mc12  007228 v12003107} & $2~10^5$ \\\hline
%$mu19$  & {\tt Single-Mu misal1\ mc12  007233 v12003107} & $2~10^5$  \\\hline
\end{tabular}
\caption{Data samples used in the muon isolation algorithm optimization. \label{tab:muiso1}}
\end{center}
\end{table}

A muon track passing through the calorimetry will deposit energy in
the cells which immediately surround it. The  deposited energy can
be contained within some cone of radius $R_{Inner}$, where $R = \sqrt{\Delta\eta^2+\Delta\phi^2}$. If the muon is isolated, there
will be little energy deposited in cells which lie in an outer
annulus around this ($R\in[R_{Inner},R_{Outer}]$). The radius of the
inner cone (i.e. the cone fully containing the muon) has been
determined from the distribution of the summed transverse energy
contained within a cone of increasing radius around the muon
direction from $Z\rightarrow\mu\mu$, as shown in
Figure~\ref{fig:muiso1}. The value of $R$ corresponding to the inner
cone radius is visible as a change in the slope of the curve. Once
the radius for which all the muon energy is contained in the cone is
reached, for each further increase of the cone radius only noise
will be summed, resulting in a reduction of the slope of the energy
sum curve. The reduction in the slope depends on the level of
electronic readout noise per cell, as shown in
Fig.~\ref{fig:muiso1}, where curves for several values of the
threshold cut on the calorimetric cell energy is shown, ranging from
40 to 90 MeV. The effect of the electronic noise is only relevant
for the LAr calorimeter. From the two figures it can be seen that a
cone of radius $0.1$ (one readout cell), is sufficient to contain
the muon energy deposition in the hadronic calorimeter, while a
radius of about $0.07$ (one to three readout cells, depending on
position), is sufficient for the electromagnetic calorimeter, due to
the finer readout granularity. The value of the outer cone radius is
instead constrained by timing requirements. Increasing the outer
cone radius requires a larger fraction of the calorimeter  to be
read out and decoded. Because the readout step of the algorithm
dominates the execution time ($>90\%$ of the overall algorithm time)
the requirement to keep the overall timing below $O(10)~ms$
constrains the maximum outer cone radius to be below about $0.4$. We
have verified that optimal background rejection is obtained by
keeping the outer cone radius at is maximum value.
% moreover
%cone radii of $0.4$ are also typically used in ATLAS for jet reconstruction both at level 1 and level 2
%of trigger, and so this value is expected to be adequate for a robust estimate of the energy released
%by the jet containing the muon.
\begin{figure}[htb!]
\begin{center}
\includegraphics[width=0.85\figwidth]{fig/muIso_et_vs_thr.eps}
\end{center}
\caption{The total transverse energy contained within a cone of increasing radius around the
muon candidate track from $Z\to\mu\mu$ signal events in the LAr calorimeter (left) and Tile calorimeter (right).
The different curves on each figure correspond to different thresholds applied on the cell energy.}
\label{fig:muiso1}
\end{figure}

An analysis has been performed over all the quantities used in the
isolation hypothesis testing, with a goal of minimizing the number
of variable used in the optimization step. Each variable used in the
optimization is listed in Table~\ref{tab:muiso2}, together with the
respective separation power expressed in term of minimum variance
bound~\cite{muiso_mvb}.
\begin{table}[h!]
\begin{center}
\begin{tabular}{|l|l|c|}\hline
Label & variable & Separation \\\hline
$var1$    & $Iso_{LAr} = {\sum E_T^{\Delta R<0.07}}/{\sum E_T^{\Delta R<0.4}}$ & 0.21  \\\hline
%$var1$    & $Iso_{LAr} = {\sum E_T^{\Delta R<0.07}}/{\sum E_T^{\Delta R<0.4}E_T}$ & 0.21  \\\hline
$var2$    & $Iso_{Tile} = {\sum E_T^{\Delta R<0.1}}/{\sum E_T^{\Delta R<0.4}}$ & 0.29  \\\hline
%$var2$    & $Iso_{Tile} = {\sum E_{\Delta R<0.1}E_T}/{\sum_{\Delta R<0.4}E_T}$ & 0.29  \\\hline
$var3$    & $E_{LAr}^{O} = \sum E_T^{\Delta R\in[0.07,0.4]}$                      & 0.75  \\\hline
$var4$    & $E_{Tile}^{O} = \sum E_T^{\Delta R\in[0.1,0.4]}$                      & 0.40  \\\hline
$var5$    & $E_{LAr}^{I} = \sum E_T^{\Delta R<0.07}$                              & 0.23  \\\hline
$var6$    & $E_{Tile}^{I} = \sum E_T^{\Delta R<0.1}$                              & 0.06  \\\hline
%$var3$    & $E_{LAr}^{O} = \sum_{\Delta R\in[0.07,0.4]}E_T$                      & 0.75  \\\hline
%$var4$    & $E_{Tile}^{O} = \sum_{\Delta R\in[0.1,0.4]}E_T$                      & 0.40  \\\hline
%$var5$    & $E_{LAr}^{I} = \sum_{\Delta R<0.07}E_T$                              & 0.23  \\\hline
%$var6$    & $E_{Tile}^{I} = \sum_{\Delta R<0.1}E_T$                              & 0.06  \\\hline
$var7$    & Number of LAr cells above threshold with $\Delta R\in[0.07,0.4]$    & 0.72  \\\hline
$var8$    & Number of Tile cells above threshold with $\Delta R\in[0.1,0.4]$    & 0.34  \\\hline
$var9$    & Number of LAr cells above threshold with $\Delta R<0.07$            & 0.31  \\\hline
$var10$   & Number of Tile cells above threshold with $\Delta R<0.1$            & 0.07  \\\hline
\end{tabular}
\caption{Variable used in the muon isolation optimization procedure. The separation is zero for identical signal and
background shapes, and it is one for shapes with no overlap.\label{tab:muiso2}}
\end{center}
\end{table}
%The correlation among the variables is reported in Fig.~\ref{fig:muiso2}.
%
%\begin{figure}[htb!]
%\begin{center}
%\includegraphics[width=0.6\textwidth]{fig/muIso_corr.eps}
%\end{center}
%\caption{Correlation matrix among the muon isolation variables used in the optimization procedure. Variable are
%labeled according to the list reported in Table~\ref{tab:muiso2}.}
%\label{fig:muiso2}
%\end{figure}
%
The optimal value of the cell energy cut thresholds, used to compute
the transverse energy and number of cell sums, has been obtained by
maximizing the background rejection after applying a fixed cut on
the isolation variables ($var1$ and $var2$ in
Table~\ref{tab:muiso2}), giving a $95\%$ efficiency for the
$Z\to\mu\mu$ signal. A common threshold value of 60 MeV has been
obtained with this procedure for both the LAr and Tile calorimeter.
Algorithm performances are stable for threshold variations of
$\pm$10 MeV around the optimal values.
%The distribution of each variable for signal and background events are shown in Figures~\ref{fig:muiso3},\ref{fig:muiso4}
%and \ref{fig:muiso5}. After the preliminary ranking analysis variable $var6$ and $var10$ have been dropped from the optimization procedure.
The distributions of some of the most powerful variables for signal
selection and background rejection  are shown in
Fig.~\ref{fig:muiso3}.

%Figure
\begin{figure}[htb]
\begin{center}
%Subfigure
%\subfigure[Number of LAr cells above threshold in the inner cone]{
%\label{F:1a}
%\includegraphics[width=0.45\textwidth]{fig/muIso_Zmumu_bb15_EM_CSUMI.eps}
%}
%Subfigure
\subfigure[Number of LAr cells above threshold in the outer ring]{
\label{F:2a}
\includegraphics[width=0.33\textwidth]{fig/muIso_Zmumu_bb15_EM_CSUMO.eps}
}
%Subfigure
%\subfigure[Transverse energy sum in the LAr in the inner cone]{
%\label{F:3a}
%\includegraphics[width=0.45\textwidth]{fig/muIso_Zmumu_bb15_EM_ESUMI.eps}
%}
%Subfigure
\subfigure[Transverse energy sum in the outer ring of LAr]{
\label{F:4a}
\includegraphics[width=0.33\textwidth]{fig/muIso_Zmumu_bb15_EM_ESUMO.eps}
}
%Subfigure
\subfigure[Transverse energy sum in the Tile in the outer ring]{
\label{F:4b}
\includegraphics[width=0.32\textwidth]{fig/muIso_Zmumu_bb15_HAD_ESUMO.eps}
}
%Subfigure
\subfigure[Isolation variable in Tile calorimeter]{
\label{F:2c}
\includegraphics[width=0.33\textwidth]{fig/muIso_Zmumu_bb15_isoHAD.eps}
}
\caption{Most powerful variables for calorimetry-based muon isolation.}
\label{fig:muiso3}
\end{center}
\end{figure}
Optimal cut values for the isolation variables described above have
been obtained in a multivariate optimization procedure by
simultaneously varying  all the cuts in sensible ranges and by
minimizing the $b\bar{b}\to\mu X$ background efficiency at fixed
$Z\to\mu\mu$ signal efficiency. In the optimization procedure, both
background and signal efficiencies are calculated with respect to
muons satisfying the L2 muFast mu20 requirement.
%After a preliminary ranking analysis, variables $var6$ and
%$var10$ were dropped from the optimization procedure.
In Fig.~\ref{fig:muiso6} the background rejection,defined as
$1/\epsilon_{BG}$ where $\epsilon_{BG}$ is the efficiency for the
$b\bar{b}$ background sample, and ($1 - \epsilon_{BG}$) as a
function of the $Z\to\mu\mu$ signal efficiency, obtained after the
optimization procedure, are shown. The chosen working point for the
isolation algorithm yields a factor of $10$ reduction for the
$b\bar{b}$ background at a $95\%$ signal efficiency for muons with
$p_T>$20~GeV.

%Figure
\begin{figure}[htb!]
\begin{center}
%Subfigure
\subfigure
%[Background rejection versus Signal efficiency]
{
%\label{F:1d}
\includegraphics[width=0.42\textwidth]{fig/muIso_Zmumu_bb15_rejVSeff_2.eps}
}
%Subfigure
\subfigure
%[$1$-background efficiency versus Signal efficiency]
{
%\label{F:2d}
\includegraphics[width=0.45\textwidth]{fig/muIso_Zmumu_bb15_rejVSeff.eps}
}
\caption{Background rejection ($1/\epsilon_{BG}$) (left), and $1-\epsilon_{BG}$ (right), as a function of
the signal efficiency as obtained in the muon isolation algorithm optimization procedure.}
\label{fig:muiso6}
\end{center}
\end{figure}

\subsection{Performance}

The performance of the isolation algorithms in terms of $b\bar{b}$
and dijet background reduction, efficiency of benchmark signal
channels and timing is presented below. The performance of isolation
algorithms can be affected by the instantaneous luminosity since the
pile-up requires higher thresholds for the same nominal efficiency.
This is particularly true for calorimetry-based isolation, while for
track-based isolation the effect can be reduced by requiring that
the contributing tracks come from the same primary vertex as the
muon. For this reason, the results of this study should be taken as
preliminary and valid only in the framework of the approximations
used in the simulated events production for these studies. Possible
changes and further development may occur as soon as real data is
available.

The effect of isolation algorithms on various sources of
non-isolated muons at L2 is shown in Table~\ref{tab:muiso3}. The
quantity $1 - \epsilon_{BG}$ is shown for the isolation requirements
corresponding to a working point for the isolation algorithm with a
nominal $Z\to\mu\mu$ signal efficiency of $95\%$. Results from dijet
decays give an estimate of the rejection power of the isolation
algorithm for high-\pt\ muons from $K$ and $\pi$  in flight decays.
The rejection power for low-\pt\ muons from $b\bar{b}$ decays
selected by the level 2 mu6 requirement has also been estimated. The
reduction in rejection power, from a factor $10$ to a factor of
about $2$, at the low-\pt\ limit is expected, given the low energy
associated with the jets. As already mentioned, calorimetry based
isolation algorithms are not effective against these kind of muons,
and the track-based isolation is expected to be much more powerful
in reducing this kind of background.

\begin{table}[htb!]
\begin{center}
\begin{tabular}{|l|c|c|c|}\hline
Process  & Trigger item & Average muon $\pt$ (GeV) & $1-\epsilon_{BG}$ (\%) \\\hline
$b\bar{b}\to\mu(15) X$    & mu20    & $25.0$  & $89.4\pm0.7$ \\\hline
$b\bar{b}\to\mu(6) X$     & mu6     & $9.0$   & $54.6\pm0.9$ \\\hline
$q\bar{q}\to\mu X$  & mu20    & $40.0$  & $99.6\pm0.1$ \\\hline
$q\bar{q}\to\mu X$  & mu6     & $20.0$  & $97.3\pm0.2$ \\\hline
\end{tabular}
\caption{Muon isolation algorithm $1-\epsilon_{BG}$ for muons from several background samples.
Efficiencies are calculated with respect to muons passing the level
2 mu20 or mu6 requirements, as specified. \label{tab:muiso3}}
\end{center}
\end{table}

The isolation algorithm has been tuned such that the efficiency  is
$95\%$ at the chosen working point. The efficiency of the isolation
requirement has been studied as a function of $\pt$ using samples of
single muons with a $\pt$ of  11, 39, and 100~GeV. No sizable change
in the muon efficiency are visible, indicating that radiation
effects are small for $\pt$ in this range. Evaluation of the effect
of the muon radiation for very high \pt\ muons (500 to 1000 GeV) is
ongoing. The efficiencies for muons from $Z\to\mu\mu$ decays and for
single muons are reported in Table~\ref{tab:muiso4}.

\begin{table}[htb!]
\begin{center}
\begin{tabular}{|l|c|c|c|}\hline
Process  & Trigger path & $\epsilon$ (\%) \\\hline
$Z\rightarrow\mu^+\mu^-$ & mu20 & $95.5\pm0.4$ \\\hline
Single $\mu~\pt$=100 GeV & mu20 & $98.68\pm0.07$ \\\hline
Single $\mu~\pt$=39 GeV & mu20 & $98.97\pm0.07$ \\\hline
Single $\mu~\pt$=6 GeV & mu6 & $98.54\pm0.09$ \\\hline
\end{tabular}
\caption{Muon isolation algorithm efficiencies for muons from several processes and thresholds.
\label{tab:muiso4}}
\end{center}
\end{table}

The time available for running L2  algorithms in the on-line trigger is limited to approximately 20~ms.
The CPU processing time is
therefore a relevant parameter for the feasibility of algorithms to be included in the trigger chain.
%A preliminary estimate of the CPU time spent by the calorimetry-based isolation algorithm has been obtained by running
%on the trigger development PC farm (pc-atr) the full muon slice over cosmic data taken during the M4 milestone run and over
%various simulated samples of signal and background events. More than 90\% of the cpu time is spent by the isolation
%algorithm accessing the calorimetric informations through the $T2CaloCommon$ access scheme. The obtained results
%algorithm accessing the calorimetric informations.
The  results obtained
%are consistent with those from  $e/gamma$ slice studies
indicate a typical overall time of less than 10~ms. Further and more
detailed timing studies performed on the actual L2 processors are
ongoing.

%\subsection{Comments and work in progress}
%The possibility to select at the ATLAS second level trigger with high efficiency isolated muons from $W$ and $Z$ decays
%reducing the ones from heavy quark decays has been studied in detail. Isolation criteria using information from
%electromagnetic and hadronic calorimeters have been developed. It must be remembered that although electronic readout noise
%and pileup noise have been simulated in the event samples used to develop such criteria, no cavern background has been
%yet included. A factor ten reduction on high \pt\ muons from heavy-quark decays has been obtained keeping $95\%$ efficiency
%on $Z\to\mu^+\mu^-$ final state.%
%
%Once estimated more realistically the CPU processing time, next step will be to investigate how much the use of the
%longitudinal granularity of the calorimeters will increase the muon isolation rejection power. Moreover, the strategy
%of using either $muComb$ or directly $muFast$ as seeding starting point need to be completed. %
%
%Isolation criteria based on the inner tracker detector to make more robust the algorithm against muon bremsstrahlung events
%and increase of the pileup with instantaneous luminosity will be presented in a forthcoming note.

\section{Muon identification using the tile calorimeter}
\label{sec:tile_perf}

The muon signatures in the three radial layers of the Tile
Calorimeter  are  well measured  quantities with a typical pattern
that can be used to  identify  the  muons efficiently down to very
low $p_{\rm T}$. This information can be used to confirm the Muon
Spectrometer Triggers (i.e. provide  redundancy in noisy/dead
regions) or to enhance the selection  efficiency for very soft muons
typically out of reach for the spectrometer.

The algorithm exploits the  radial and transverse calorimeter
segmentation.  The search starts from the outermost layer, which is
the one with the cleanest signal, and once a cell is found with
energy compatible with a muon, the algorithm checks the energy
deposition in the neighbor cells for the most internal layers. These
candidate patterns'' are considered as  muons when cells
compatible with the typical muon energy deposition are found
following a $\eta$-projective pattern in all the three TileCal
layers. More details can be found in Ref.~\cite{TileMuId}.

\subsection{Performance}
\label{subsec:TileMuId}

The performance of the TileMuId algorithms has been studied with
MonteCarlo  single muons and semi-inclusive muon production
($b\bar{b}\rightarrow\mu(4)X$) samples.
The effect of minimum-bias pileup at low luminosity ($\mathcal{L}=10^{33} {\rm cm}^{-2}{\rm s}^{-1}$) has been investigated as well.

Two algorithms, implementing complementary strategies are described,
one (TrigTileLookForMuAlg) is  fully executed on the LVL2 Processing
Unit (PU) the second (TrigTileRODMuAlg) has a core part executed on
the Read Out Driver Digital Signal Processor (ROD-DSP)  in order to
save time. This allows  a  very fast  processing of the entire
detector (full scan) as  opposed to the RoI based processing typical
of the trigger algorithm running on the LVL2 PUs. Since each ROD-DSP
processes a small part of the detector readout the TrigTileRODMuAlg
acceptance is lower compared with that of TrigTileLookForMuAlg.

\subsubsection{Spatial resolution}

The spatial resolution of the algorithms can be  studied using the
distributions of the residuals $\Delta\eta =\eta(\mu_{\rm Tile}) - \eta(\mu_{\rm Truth})$ and $\Delta\phi=\phi(\mu_{\rm Tile}) - \phi(\mu_{\rm Truth})$ in single muon events with $2$ $\le p_{\rm T} \le 15$ GeV.
%Figure~\ref{fig:TileMuId_resolution} shows these distributions for two
%algorithms, TrigTileLookForMuAlg and TrigTileRODMuAlg.
%\begin{figure}[htb!]
%\begin{center}
%\subfigure[Distribution of residuals $\Delta\eta$ for TrigTileLookForMuAlg.]
%{\includegraphics[height=50mm]{fig/TileMuId_Lookdeta.eps}}\qquad
%\subfigure[Distribution of residuals $\Delta\phi^{\rm TR}$ for TrigTileLookForMuAlg.]
%{\includegraphics[height=50mm]{fig/TileMuId_Lookdphi.eps}}
%\subfigure[Distribution of residuals in $\Delta\eta$ for TrigTileRODMuAlg.]
%{\includegraphics[height=50mm]{fig/TileMuId_RODdeta.eps}}\qquad
%\subfigure[Distribution of residuals in $\Delta\phi^{\rm TR}$ for TrigTileRODMuAlg.]
%{\includegraphics[height=50mm]{fig/TileMuId_RODdphi.eps}}
%\caption{Distribution of residuals between the coordinates of the muons
%identified
%by both TileMuId algorithms and the truth muons in single muon events.}
%\label{fig:TileMuId_resolution}
%\end{center}
%\end{figure}
%In order to compare the reconstructed $\phi$-coordinates
%with the Monte Carlo truth  that is defined at the
%vertex,  the latter is extrapolated at the TileCal Radius  using the following
%parametrization:
%
%\phi ^{\rm TR} (\mu^{\pm}_{\rm Truth})= \phi(\mu^{\pm}_{\rm Truth}) \mp 0.000123 \mp \frac{0.507}{p^{\rm Truth}_{\rm T}}
%\label{eq:Phi_TrktoCal}
%
%where $p_{\rm T}$ is expressed in GeV and $\phi$ in rad. Therefore,
%the average shift in the residual $<\Delta\phi ^{\rm TR}>$
%using $\phi ^{\rm TR}$
%at the different $p_{\rm T}$ is canceled.
%Note that the $\Delta\eta$-distribution for TrigTileLookForMuAlg is slightly
%biased toward positive values due to an
%unexpected  feature of the algorithm, that results in an asymmetry between
%the positive and negative
%side. The muons tagged at $|\eta| \simeq 1.4$ are split among two search
%paths due to the coarse
%granularity of the detector and the lack of projectivity in the segmentation.
%Since the direction''
%in the detector scan is fixed from the negative to the positive sense in
%$\eta$, we pick up mostly one or the other with different paths in two
%detector-partition sides.
%The $\Delta\eta$-distributions are fitted with a Gaussian.
The distributions are well described by a Gaussian and resolution
can be defined as  $\sigma_{\eta}$ = 0.05 for TrigTileRODMuAlg (
$\sigma_{\eta}$ = 0.04 for TrigTileLookForMu) and $\sigma_{\phi}$ =
0.03~rad. To characterize the performance of the algorithms with MC
physics events a matching region with the MC truth will be used. For
this analysis a matching region of
$\Delta\eta\times\Delta\phi=0.2\times0.12$ is used.
%It leads the efficiency of TileLookForMu to increase up to 0.5 \%.
%and  about XX\% for the fraction of fakes.

\subsubsection{Efficiency}

The muon-tagging efficiency is defined as the ratio of the
number of tagged muons which match a truth muon ($N_{\rm tag}$)
to the number of generated truth muons ($N_{\rm gen}$).
%
%\epsilon = \frac{N_{\rm tag}}{N_{\rm gen}}.
%
\begin{figure}[htb!]
\begin{center}
%\subfigure{\includegraphics[height=50mm]{fig/TileMuId_singlemueeta.eps}}
%\subfigure{\includegraphics[height=50mm]{fig/TileMuId_singlemuephi.eps}}
%\subfigure{\includegraphics[height=50mm]{fig/TileMuId_singlemuept.eps}}
\includegraphics[height=50mm]{fig/TileMuId_singlemueeta.eps}
\includegraphics[height=50mm]{fig/TileMuId_singlemuephi.eps}
\includegraphics[height=50mm]{fig/TileMuId_singlemuept.eps}
\caption{Efficiency as a function of $\eta$ (left), $\phi$ (center) and $p_{\rm T}$ (right)for
TrigTileLookForMu (filled circles) and TrigTileRODMu (open squares)
using single muon events.}
\label{fig:single_muons_eff_eta_phi_pt_Look_ROD_tight}
\end{center}
\end{figure}
\begin{figure}[htb!]
\begin{center}
\subfigure{\includegraphics[height=50mm]{fig/TileMuId_mu6eeta.eps}}
%\subfigure{\includegraphics[height=50mm]{fig/TileMuId_mu6ephi.eps}}
%\subfigure{\includegraphics[height=50mm]{fig/TileMuId_mu6ephi.eps}}
\subfigure{\includegraphics[height=50mm]{fig/TileMuId_mu6ept.eps}}
\subfigure{\includegraphics[height=50mm]{fig/Eff_TrueTile.vsPt.bbmu6X.Test.F3.eps}}
\caption{Efficiency  as a function of $\eta$ (left) and $p_{\rm T}$ (center)
for TrigTileLookForMu (filled circles) and TrigTileRODMu (open squares)
in $b\bar{b}\rightarrow\mu(6)X$  events. Right plot show for TrigTileLookforMu
the effect  of pileup of  Minimum Bias events at low luminosity
(filled circles) compared with the case without pileup  (open squares).}
\label{fig:bbmu6X_eff_eta_pt_Look_ROD_tight}
\end{center}
\end{figure}
Figure~\ref{fig:single_muons_eff_eta_phi_pt_Look_ROD_tight} shows
the efficiency as a function of $\eta$, $\phi$, and $p_{\rm T}$ of
the muon for the two algorithms as obtained using the  single muon
sample. The efficiency of  TileRODMu is lower than that of
TileLookForMu. In the region $0.8 \le |\eta| \le 1.1$ the towers are
split between the barrel  and the extended barrels, and the cells
belonging to different  partitions  are processed by different ROD
DSPs. Similar  effects are observed  for the boundary at $\eta\sim 0$.
%At the ROD level the muon tagging algorithm does not cover completely
%these particular towers.
%The lower efficiency from TileRODMu is also observed for $\eta\sim 0$,
%as the central cell in the outermost layer (D0 cell) is read-out by
%two photomultipliers that are connected to different readout system
%and finally allow their signal to be processed by different ROD DSPs.
Except for these two regions of low geometrical acceptance both algorithms show
efficiency $\sim 85\%$ with good agreement.
Since TileCal is homogeneous in $\phi$, the efficiency is uniform as
a function of $\phi$, see
Fig.~\ref{fig:single_muons_eff_eta_phi_pt_Look_ROD_tight} (center).
The efficiency decreases with  the muon $p_{\rm T}$ for $p_{\rm T}<3$~GeV and is about 42$\%$ at $p_{\rm T}=2$~GeV. Most of the
muons with $\pt < 2$~GeV stop in the Tile
calorimeter.
For $\pt > 4$~GeV  the efficiency is  flat  at about 60$\%$.
Figure~\ref{fig:bbmu6X_eff_eta_pt_Look_ROD_tight} (left) and
(center) show the efficiency curves for  both algorithms as obtained
in $b\bar{b}\rightarrow\mu(6)X$ events. These results are in good
agreement with the performance obtained using single muons,
indicating that the algorithms are not too sensitive to the
additional hadronic activity in $b\bar{b}$ events.
%Similary to the single muons case, large differences between
%the efficiency of the two algorithms can be seen  in the gap region.

%The previous results are valid in the limit of very low luminosity
%when the pileup of minimum-bias events can be neglected.
To  evaluate the performance in  a  realistic LHC operation scenario
a sample of $b\bar{b}\rightarrow\mu(6)X$ events simulated with
pileup of minimum-bias  events  at a luminosity
$\mathcal{L}=10^{33}$ cm$^{-2}$s$^{-1}$ was used. As shown in
Fig.~\ref{fig:bbmu6X_eff_eta_pt_Look_ROD_tight} (right), the
efficiencies as a function of $p_{\rm T}$ for two cases are similar
for $p_{\rm T}> 5$ GeV. The additional muons from minimum-bias
events make the efficiency worse in the low $p_{\rm T}$ region. The
average efficiency in the  sample with pileup (67.97$\pm$0.81)$\%$
is slightly lower than the one obtained without pileup
(74.25$\pm$0.79)$\%$. It can be concluded that the efficiency is not
substantially affected by  minimum-bias pileup.

%\begin{figure}[htb!]
%\begin{center}
%\subfigure{\includegraphics[height=50mm]{fig/Eff_TrueTileTrk.vsEtaT.bbmu6X.Test.F3.eps}}
%\subfigure{\includegraphics[height=50mm]{fig/Eff_TrueTileTrk.vsPhiT.bbmu6X.Test.F3.eps}}
%\subfigure{\includegraphics[height=50mm]{fig/Eff_TrueTile.vsPt.bbmu6X.Test.F3.eps}}
%%\subfigure{\includegraphics[height=50mm]{fig/TileMuId_mu6pileeeta.eps}}\qquad
%%\subfigure{\includegraphics[height=50mm]{fig/TileMuId_mu6pileept.eps}}
%\caption{TrigTileLookForMuAlg efficiency (tight selection) as a function of $\eta$ and $p_{\rm T}$ for $b\bar{b}\rightarrow\mu(6)X$ events
%with (filled circles) and without pileup (open squares).}
%\label{fig:bbmu6X_pileup_eff_eta_pt_Look_tight}
%\end{center}
%\end{figure}

\subsubsection{Fraction of fakes}

\begin{figure}[htb!]
\begin{center}
\includegraphics[height=50mm]{fig/TileMuId_mu6feta.eps}
\includegraphics[height=50mm]{fig/TileMuId_mu6fphi.eps}
\includegraphics[height=50mm]{fig/ffake_TrueTile.vsEtaM.bbmu6X.Test.F3.eps}
\caption{Fraction of fakes  as a function of $\eta$ (left) and
$\phi$ (center) for TrigTileLookForMu (filled circles) and for
TrigTileRODMu (open squares) in $b\bar{b}\rightarrow\mu(6)X$ events.
The right plot compare performance of  TrigTileLookForMu in samples with
(filled circles) and  without pileup (open squares).}
\label{fig:bbmu6X_ff_eta_phi_Look_ROD_tight}
\end{center}
\end{figure}
The muon tags which are not matched with  truth muons
are considered fake.  The same  $\Delta \eta \times \Delta\phi$
matching cuts  are used for the efficiency and fake computation.
The fraction of fakes in a given  data sample is defined  as the ratio
of the number of misidentified muons to the total number of
events.

The left and center plots of
Fig.~\ref{fig:bbmu6X_ff_eta_phi_Look_ROD_tight} show the fraction of
fakes as a function of $\eta$ and $\phi$
%$\frac{1}{N_{\rm event}} \frac{dN_{\rm f}}{d\eta}$
%and equivalent distribution, $\frac{1}{N_{\rm event}}\frac{dN_{\rm f}}{d\phi}$
obtained by the two algorithms. Both algorithms show a very small
fake rate in the central region  (0.12$\%$  for  $|\eta|<0.7$). The
main contribution  of fakes comes from the extended barrel and gap
regions,  where the cell segmentation is coarse and the projectivity
is the worst. The fraction of fakes in the whole range $|\eta|<1.4$
is $2.7 \pm 0.1$\% for TrigTileRODMu and $4.1 \pm 0.1$\% for
TrigTileLookForMu. The fraction of misidentified muons as a function
of $\phi$ is flat as expected.

Figure~\ref{fig:bbmu6X_ff_eta_phi_Look_ROD_tight} (right) shows the
performance  of TrigTileLookForMu in  $b\bar{b}\rightarrow\mu(6)X$
events with  and without the pileup of minimum-bias events at low
luminosity. The  fraction of fakes increase from 3.7 $\pm$ 0.1\% to
6.0 $\pm$ 0.1\% when the  minimum bias pileup at
$\mathcal{L}=10^{33}$ cm$^{-2}$s$^{-1}$  is taken into account. The
fake rate increases at larger values of $\eta$ (gap and extended
barrel), where the cell granularity is worse and more minimum bias
event are expected, compared to the central $\eta$ region.

%Note that the fraction of fakes and the efficiency can be tuned
%changing  the cell lower energy thresholds~\cite{TileMuId}.
%This study was performed with all the thresholds set to 80~MeV.
%
%\begin{figure}[htb!]
%\begin{center}
%\subfigure{\includegraphics[height=50mm]{fig/TileMuId_mu6pilefeta.eps}}\qquad
%\subfigure{\includegraphics[height=50mm]{fig/TileMuId_mu6pilefphi.eps}}
%\caption{ Differential distributions  of  fakes (TrigTileLookForMuAlg)
%as a function of $\eta$ and $\phi$ in  $b\bar{b}\rightarrow\mu(6)X$ events
%simulated with (open) and without (full) pileup of Minimum bias events.}
%\label{fig:bbmu6X_pileup_ff_eta_phi_Look_tight}
%\end{center}
%\end{figure}

%Table~\ref{tab:comparison_Look_ROD} summarizes the average
%efficiencies  and fraction of fakes found in
%$b\bar{b}\rightarrow\mu(6)X$ process using  both algorithms
%and the impact of minimum-bias pileup on their performance.
%
%\begin{table}[htb!]
%\begin{center}
%\begin{tabular}{|l|c|c|c|c|}
%\hline
% & \multicolumn{2}{|c|}{\bf Efficiency (\%)} & \multicolumn{2}{|c|}{\bf Fraction of fakes (\%)} \\
% & {\bf without pileup} & {\bf with pileup} & {\bf without pileup} & {\bf with pileup} \\
%\hline
%TileLookForMu & 71.8  $\pm$ 0.4  & 67.97 $\pm$ 0.81
%                     &  4.08 $\pm$ 0.14 &  6.04 $\pm$ 0.14 \\
%
%TileRODMu     & 56.9  $\pm$ 0.4  &
%                     &  2.74 $\pm$ 0.11 & \\
%\hline
%\end{tabular}
%\end{center}
%\caption{Average efficiencies and fraction of fakes for TileMuId algorithms
%for $b\bar{b}\rightarrow\mu(6)X$.}
%\label{tab:comparison_Look_ROD}
%\end{table}

\subsection{Combined performance with the inner detector}

In order to measure the $\pt$ of the identified muon, the
secondary RoI produced by the TileMuId algorithm
is used to seed the Inner Detector (ID) track reconstruction algorithm.
%TrigIDSCAN~\cite{IDScan-1, IDScan-2}.
The size of the ID RoI that requires processing  is defined  by the
Tile algorithm resolution and by the bending in the central
solenoid.
%Essentially  the $\Delta \eta = \eta(\mu_{\rm Tile}) - \eta(\mu_{\rm Track})$
%is defined by the resolution plots in Fig.~\ref{fig:TileMuId_resolution}.
%\begin{figure}[htb!]
%\begin{center}
%\includegraphics[width=.9\textwidth]{fig/TileMuId_Pl_DCOrPhiPerPt.NewCut.TileTrk.eps}
%\end{center}
%\caption[Difference of $\phi$ between the muon tagged by Tile Calorimeter and the matched track.]
%        {Difference of $\phi$ between the muon tagged by Tile Calorimeter and the
%        matched track before(left)/after(right) applying the extrapolation in
%        the Tile Calorimeter position.
%        The lines indicate the average-value of
%        $|\phi(\mu_{\rm Tile}) - \phi(\mu_{\rm Track})|$ in each bin.
%        The events with positive (negative) sign of
%        $\phi(\mu_{\rm Tile}) - \phi(\mu_{\rm Track})$ are from $\mu ^-$
%        ($\mu ^+$).}
%\%label{fig:DCorPhi_TileTrk}
%\end{figure}
%Figure~\ref{fig:DCorPhi_TileTrk} shows the
%$\Delta \phi = \phi(\mu_{\rm Tile}) - \phi(\mu_{\rm Track})$ as
%a function of truth $p_{\rm T}$  for muons tagged by the algorithm.
%Since the charge is not  known we need to consider both directions.
For $p_{\rm T}(\mu_{\rm Truth}) >2$ GeV, $\Delta \phi = \phi(\mu_{\rm Tile}) - \phi(\mu_{\rm Track}) \approx 0.2$ is
required.
% as shown in Fig.~\ref{fig:DCorPhi_TileTrk}.
If at least one track is found  within the region $\Delta \eta \times \Delta \phi= 0.1 \times 0.2$ and with $p_{\rm T}>2$ GeV, the
calorimetric tag is confirmed  to be a muon and the trigger sequence
is successful.
%The $\phi$ of track  is  extrapolated at  the Tile Calorimeter radius.
%using  equation (\ref{eq:Phi_TrktoCal}).
%The extrapolated $\phi ^{\rm TR} (\mu _{\rm Track})$ is very close to the
%value of $\phi (\mu_{\rm Tile})$ (see Fig.~\ref{fig:DCorPhi_TileTrk}).
\begin{figure}[thb!]
\begin{center}
\includegraphics[height=50mm]{fig/NtrkRoI_TrueTile.bbmu4X.F102.eps}
\includegraphics[height=50mm]{fig/NtrkRoI_TrueTile.Pilewo.bbmu6X.Test.F102.eps}
\includegraphics[height=50mm]{fig/Eff_TrueTileTrk.vsPt.ITests.3.bbmu4X.F102.eps}
\end{center}
\caption[Number of tracks within the given RoI size and efficiency as a function of $p_{\rm T}$]
{The number of tracks within the given RoI  for
$b\bar{b} \to \mu(4)X$ (left) and with the
%RoI size $(\Delta \eta, \Delta \phi)=(0.1,0.2)$ for
fixed RoI size of $\Delta \eta =0.1$ and $\Delta \phi=0.2$ for
$b\bar{b} \to \mu(6)X$ with/without pileup (center).
The right plot shows the efficiency as a function of $p_{\rm T}$
for the muons tagged by only TileCal (TileLookForMu)
and the muons combined with
the associated track.}
\label{fig:Eff_TruthTileTrk}
\end{figure}

Figure~\ref{fig:Eff_TruthTileTrk} shows the multiplicity of track in
the  ID RoI; left plot  shows that a region with $\Delta\phi = 0.1$
misses the low $p_{\rm T}$ tracks and results in more  events  with
zero track within the RoI. The RoI  with $\Delta\eta = 0.2$ does not
give any advantage. The RoI  with  a size $\Delta \eta =0.1$ and
$\Delta \phi=0.2$  is a good compromise; the  efficiency to
reconstruct the muon track is good and the  multiplicity of tracks
(ambiguity) is acceptable. In the case of  reconstruction of
multiple tracks,  the closest  is chosen as  the best-matched for
the $\mu$ tagged by TileCal, and all tracks are saved since the
ambiguity cannot be further resolved at this level. As shown in
Figure~\ref{fig:Eff_TruthTileTrk} (center), the multiplicity  of
tracks within the RoI is  not significantly affected by the pileup.

Figure~\ref{fig:Eff_TruthTileTrk} (right) shows  the overall
combined (TileCal+ID) efficiency for  $\Delta \phi = 0.1$  and
$\Delta \phi = 0.2$ as a function of muon $\pt$. The combined
efficiency obtained with $\Delta \phi = 0.2$ is approximately  equal
to  that of the TileCal  stand-alone  except for $p_{\rm T} < 3.5$
GeV. The efficiency from the matched track shows no dependence on
$\eta$ or $\phi$. The efficiency, purity and acceptance using the
different sizes of RoI are summarized in
Table~\ref{tab:eff_roisize}. The efficiency and acceptance are
significantly improved from $\Delta \phi = 0.1$ to $\Delta \phi = 0.2$. For $\Delta \phi = 0.2$,  97\% of tagged muons by TileCal
match the associated track. The purity and acceptance of $\Delta \phi = 0.3$ are similar to those of $\Delta \phi = 0.2$. However,
the size of $\Delta \eta$ does not affect the efficiency of the
matched track with $\mu$ significantly. The differences due to the
minimum-bias  pileup are observed to be about 2 to 3\% due to the
small number of events from pileup samples.

\begin{table}[htb!]
\begin{center}
\begin{tabular}{|l|c|c|c|c|}
\hline
& {\bf TileLookForMu} & \multicolumn{3}{|c|}{\bf RoI size with ($\Delta\eta$, $\Delta\phi$) for matching tracks  } \\
&            &(0.1, 0.1) & (0.1, 0.2) & (0.1, 0.3) \\
\hline
%\multicolumn{5}{|l|}{\bf $b\bar{b}\rightarrow\mu(4)X$} \\
\hline
Efficiency ($\%$)                 & 73.08 $\pm$ 0.17 & 42.02 $\pm$ 0.19
& 70.91 $\pm$ 0.18 & 72.06 $\pm$ 0.18 \\
Unmatched $\mu_{\rm Tile}$ ($\%$) &                  & 42.50 $\pm$ 0.36
&  2.98 $\pm$ 0.08 & 1.40  $\pm$ 0.05 \\
Efficiency ($p_{\rm T} > 4$ GeV)  &                  & 44.09 $\pm$ 0.20
& 72.94 $\pm$ 0.18 & 73.06 $\pm$ 0.18 \\
Purity ($p_{\rm T} > 4$ GeV )      &                  & 98.51 $\pm$ 0.88
& 98.79 $\pm$ 0.67 & 98.69 $\pm$ 0.67 \\
Acceptance ($p_{\rm T} > 4$ GeV)  &                  & 40.78 $\pm$ 0.31                                           & 71.93 $\pm$ 0.45 & 72.01 $\pm$ 0.45 \\

%\hline
% & {\bf TrigTileLookForMu} & \multicolumn{3}{|c|}{\bf RoI size with ($\Delta\eta$, $\Delta\phi$) for matching tracks  } \\
% & loose selection & (0.1, 0.1) & (0.1, 0.2) & (0.1, 0.3) \\
%\hline
%Efficiency ($\%$)                 & 79.55 $\pm$ 0.36 & 46.37 $\pm$ 0.34 & 77.22 $\pm$ 0.35 & 78.46 $\pm$ 0.36 \\
%Unmatched $\mu_{\rm Tile}$ ($\%$) &                  & 41.71 $\pm$ 0.38 &  2.93 $\pm$ 0.43 &  1.37 $\pm$ 0.43 \\
%Efficiency ($p_{\rm T} > 4$ GeV)  &                  & 48.84 $\pm$ 0.35 & 79.76 $\pm$ 0.37 & 79.91 $\pm$ 0.37 \\
%Purity ($p_{\rm T} > 4$ GeV)      &                  & 97.67 $\pm$ 0.59 & 98.17 $\pm$ 0.45 & 98.05 $\pm$ 0.45 \\
%Acceptance ($p_{\rm T} > 4$ GeV)  &                  & 45.67 $\pm$ 0.35 & 79.14 $\pm$ 0.37 & 79.24 $\pm$ 0.37 \\
\hline
\end{tabular}
\end{center}
\caption{Performance with the matched track for $b \bar{b} \to \mu(4)X$.}
\label{tab:eff_roisize}
\end{table}

%In order to  reduce the processing time to reconstruct the track
%within the given RoI size and also to reduce the ambiguity
%from the multiplicity of tracks, $\Delta \eta = 0.1$ and $\Delta \phi = 0.2$
%are chosen.
%The TrigTileMuFeX$\_$L2 algorithm inside TrigTileMuId package
%compare the extracted tracks from Inner Detector with the
%tagged muons from Tile Calorimeter and  store the best-matched track(s)
%in the Feature, TileTrkMuFeature''.
%This  will be used in the TileMuHypo algorithm under
%TrigMuonHypo package.

%\subsection{Conclusions}

%The overall performance of the Tile muon tagging algorithm has
%been presented in this section for both implementations of the
%algorithm  (TrigTileLookForMuAlg and TrigTileRODMuAlg) and the
%two selections defined (tight and loose)
%using MC samples of single muons and inclusive B-Physics processes,
%including  minimum-bias pileup at low luminosity.

\section{Muon trigger performance for \Zmumu
%$Z \rightarrow \mu\mu$
}
\label{sec:trig_from_data}

\subsection{The tag and probe'' method}

\begin{figure}[b!]
\centering
\begin{minipage}[c]{0.45\textwidth}
\includegraphics[width=0.9\textwidth]{FiguresBellomo/TagProbeIllustration.eps}
\end{minipage}
\begin{minipage}[c]{0.45\textwidth}
\footnotesize
\begin{tabular}{|lll|}
\hline
\textsf{\textbf{Process}}   & \textsf{\textbf{Generation cuts}} & \boldmath{$\sigma$} \textsf{\textbf{[pb]}}  \\
\hline
\hline
$Z \rightarrow \mu^+\mu^-$                                          & $M_{\mu\mu} >$ 60~GeV/$c^2$                                     & 1497 \\
& 1$\mu$: $|\eta| < 2.8, p_{T} >$ 5~GeV                 &  \\
\hline
\hline
%$W \rightarrow \mu\nu$
$\Wmn$                                                &1$\mu$: $|\eta| < 2.8, p_{T} >$ 5~GeV                   & 11946 \\
& & \\
\hline
$BB \rightarrow \mu\mu X$                                           & 1$\mu$: $|\eta| < 2.5, p_{T} >$ 15~GeV                & 4000 \\
& 1$\mu$: $|\eta| < 2.5, p_{T} >$ 5~GeV                  &  \\
\hline
$t\bar{t} \rightarrow W^{+}bW^{-}b$                              & only leptonic decay                          & 461 \\
\hline
$Z \rightarrow \tau^+\tau^-$                                         &   $\tau\tau \rightarrow ll, M_{\mu\mu} >$ 60~GeV/$c^2$ & 77 \\
& 1$\mu$: $|\eta| < 2.8, p_{T} >$ 5~GeV                  &  \\
\hline
\end{tabular}
\end{minipage}
\caption{Illustration of the Tag and Probe method (left) and
cross-sections with generation cuts for signal and background
processes (right).} \label{TaP_Muon_Trigger_Samples}
\end{figure}
The trigger efficiency is a fundamental parameter in physics
analyses and therefore it is important to have several independent
methods for estimating it. The Tag and Probe'' method is a
concrete application of a data-driven technique for performance
analysis. This method is based on the definition of a probe-like''
object, used to make the performance measurement, within a properly
tagged'' sample of events. Physics processes suitable for this
method are generally those characterized by a double-object final
state signature. The decay of the Z provides two high-$p_{T}$ muons
that can lead to two trigger tracks in the
Inner Detector and Muon Spectrometer and to a  combined object.
These two measurements are in principle independent, thought not
necessarily uncorrelated.

Tagged'' events require one triggered track with $\pt~>$ 20~GeV
%and, within this sample, the
and Probe'' objects can be defined as \emph{Inner Detector offline
reconstructed tracks (ID-Probe)}, where measurements are referred to
the offline Inner Detector reconstruction efficiency ($\sim$100\%),
or as \emph{Muon Spectrometer offline reconstructed tracks
(MS-Probe)}, where values are  normalized to the offline Muon
Spectrometer reconstruction efficiency (standalone or combined with
the Inner Detector).

The trigger performance is measured by checking for L1, L2, and EF
trigger tracks associated with each probe object. A schematic
illustration of the method is shown in
Fig.~\ref{TaP_Muon_Trigger_Samples}. It must be verified that
selected tracks come from a Z decay. A background process with two
isolated tracks in the Inner Detector, of which only one is a real
muon, would introduce a systematic error in the efficiency
evaluation.
For this reason, cuts have to be applied in order to select a clean signal sample.\\
A significant background contribution is expected from QCD
processes, which have large cross-sections. This background has been
studied by considering the dominant contribution of muons from
decays of $B$-meson pairs. Also the muonic W boson decay, which can
give a higher energetic muon plus an additional muon from a QCD jet
and the $Z \rightarrow \tau^{+}\tau^{-} \rightarrow \mu^{+}\nu_{\mu}\bar{\nu_{\tau}}~\mu^{-}\bar{\nu_{\mu}}\nu_{\tau}$
process have been considered.

Moreover the top-pair production cross-section at LHC is of the same
order of magnitude as $Z$ boson cross-section. Top quarks decay with
a 99.9\% probability into a $W$boson  and a $b$ quark. Therefore
muons originating from $W$ boson and $b$-quark decays can also give
a signal-like signature. Cross-sections and generation cuts of the
processes considered are reported  in
Fig.~\ref{TaP_Muon_Trigger_Samples}. PYTHIA~\cite{pythia5} is used
to generate the processes.

Another possible source of background is muons from cosmic-rays.
%%%%%%%%%%%%%%%%
%\begin{figure}[t!]
%\centering
%\begin{minipage}[c]{0.5\textwidth}
%\footnotesize
%\begin{tabular}{|c|c|}
%\hline
%\textbf{Cut on}            &   \textbf{Requirement}                    \\
%\hline
%\hline
%Charge             &   opposite                    \\
%\hline
%Invariant Mass Requirement &   $|91.2\,GeV - M_{\mu \mu}^{rec}|<10\,GeV$   \\
%\hline
%Transverse Momentum \pt\  &   $>20\,GeV$                    \\
%\hline
%$N^{ID}_{0.05<r<0.5}$  &   $\leq4$             \\
%\hline
%$\sum_{0.05<r<0.5} p_T^{ID Tracks}$ & $\leq8\,GeV$       \\
%\hline
%$\sum_{0.05<r<0.5} E_{T}$ &    $\leq6\,GeV$      \\
%\hline
%$E_{r<0.5}^{Jet}$ &    $\leq15\,GeV$ \\
%\hline
%\end{tabular}
%\end{minipage}
%\begin{minipage}[c]{0.4\textwidth}
%\includegraphics[width=1.1\textwidth]{FiguresBellomo/Zmumu_MuonTriggerEfficiency_TP_CutFLow_mu20_IDProbe_thr_HighLumi.eps}
%\end{minipage}
%\caption{Selection cuts (left) and the cut-flow diagram for ID-probe and for an integrated luminosity of 50~pb$^{-1}$. L1, L2 and EF cuts mean that a trigger track can be associated to the probe one.}
%\label{TaP_Cuts}
%\end{figure}
%%%%%%%%%%%%%%%%
An estimation of cosmic rates in the trigger system has been done in
Ref.~\cite{AtlasLVL1TDR} and shows a negligible  effect on trigger
performance.

The isolation variables
%, defined in a hollow cone around the candidate muon
%\footnote{The isolation cone is defined in the $\eta-\phi$ plane of the candidate track as:
%\begin{eqnarray}
%r_{1} < \sqrt{(\eta_{\mu} - \eta_{ic})^{2} + (\phi_{\mu} - \phi_{ic})^{2}} < r_{2}
%\end{eqnarray}
%where the index $ic$ stands for the in-cone tracks.
%The internal radius cone $r_{1} = 0.05$ is choosen to exclude the candidate track itself from the calculation.
%The outer radius $r_{2}$ is set to 0.5.},
chosen for this analysis are the number of reconstructed tracks in the Inner Detector ($N^{ID}_{cone}$),
the sum of $\pt$ of the Inner Detector tracks ($\sum p^{ID}_{T,cone}$),
the energy of a jet candidate ($E^{jet}_{cone}$) and
the sum of reconstructed energy in the cells of the Calorimeter ($\sum E^{EM}_{cone}$).
%\begin{itemize}
%\item number of reconstructed tracks in the Inner Detector ($N^{ID}_{cone}$);
%\item sum of transverse momentum of Inner Detector tracks ($\sum p^{ID}_{T,cone}$);
%\item energy of a possible reconstructed jet ($E^{jet}_{cone}$);
%\item sum of reconstructed energy in the cells of the Calorimeter ($\sum E^{EM}_{cone}$).
%\end{itemize}
Muons from QCD processes tend to be produced within a large cascade
of other particles and therefore should not appear isolated in the
detector.  In the case of the decay of top pairs, one highly
energetic and isolated muon can come from one $W$ boson decay while
the second $W$ boson  can decay leptonically into a high-$\pt$
electron which appears as an isolated track in the Inner Detector.
In order to not count this as a false probe, electrons are vetoed.
The values of the selection cuts applied in this analysis have been
defined
%accordingly to other studies performed inside ATLAS Standard Model group
in~\cite{WZXSecCSCNote}.
% and are reported in Table \ref{TaP_Cuts}.
The isolation cuts allow for background rejection of approximately 99\% while retaining a signal efficiency of about 76\%.
%The cut-flow diagram of the probe selection is shown in Figures \ref{TaP_Cuts}.
%The background suppression is clearly visible:
After applying the probe selection cuts the signal to background ratio is more than $10^3$.
In addition, probe muons selected from background processes can be associated to trigger tracks
and hence have no negative impact on trigger efficiency measurements.

\subsection{Determination of trigger efficiencies}

Two measurement scenarios have been studied:
\begin{itemize}
\item \textit{Low luminosity} ($\int\mathcal{L}dt \simeq$ 50~pb$^{-1}$): in order to  not rely on the combined reconstruction based on Inner Detector and Muon Spectrometer matching, only the tracks from the Muon Spectrometer are used. The isolation cuts are also based only on Inner Detector quantities;
\item \textit{High luminosity} ($\int\mathcal{L}dt \simeq 1000~pb^{-1}$): full combined information from Inner Detector and Muon Spectrometer is used and also Calorimeter based cuts are applied to select isolated tracks.
\end{itemize}
In each scenario both the ID- and MS-Probe methods have been studied.
The efficiency dependence on $\phi$ and $\eta$ is determined by the Muon Spectrometer layout.
%and the binning in these quantities has been chosen accordingly to it.
A $\pt$ cut of 20~GeV   has been applied on the probe tracks to test
the system in its plateau region. The efficiency as a function of
p$_{T}$ has been also estimated from data in the \emph{high
luminosity} scenario.
%%%%%%%%%%%%%%%%
%\begin{figure}[t]
%\begin{center}
%\includegraphics[width=0.49\textwidth]{FiguresBellomo/Zmumu_MuonTrigger_EfficiencyDiff_mu20_3levels_IDProbe_thr2_LowLumi_eta.eps}
%\includegraphics[width=0.49\textwidth]{FiguresBellomo/Zmumu_MuonTrigger_EfficiencyDiff_mu20_3levels_IDProbe_thr2_LowLumi_phi.eps}
%\caption{Fractional efficiency difference of each trigger level as a function of $\eta$ and $\phi$ in the \textit{low luminosity}  scenario using the ID-Probe.}
%\label{TaP_Muon_Trigger_LowLumi_Diff_IDProbe}
%\end{center}
%\end{figure}
%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.43\textwidth]{FiguresBellomo/Zmumu_MuonTrigger_Efficiency_mu20_3levels_IDProbe_thr2_LowLumi_eta.eps}
%\includegraphics[width=0.49\textwidth]{FiguresBellomo/Zmumu_MuonTrigger_Efficiency_mu20_3levels_IDProbe_thr2_LowLumi_phi.eps}
\includegraphics[width=0.43\textwidth]{FiguresBellomo/Zmumu_MuonTrigger_EfficiencyDiff_mu20_3levels_IDProbe_thr2_LowLumi_eta.eps}
\caption{The muon trigger efficiency for each trigger level (left) and fractional efficiency difference (right) as a function of $\eta$
in the \textit{low luminosity}  scenario using the ID-Probe.
The efficiencies determined with the Tag and Probe method are compared to those calculated in a Monte Carlo truth-based analysis.}
\label{TaP_Muon_Trigger_LowLumi_Eff_IDProbe}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.43\textwidth]{FiguresBellomo/Zmumu_MuonTrigger_Efficiency_mu20_3levels_MSProbe_thr2_LowLumi_eta.eps}
\includegraphics[width=0.43\textwidth]{FiguresBellomo/Zmumu_MuonTrigger_EfficiencyDiff_mu20_3levels_MSProbe_thr2_LowLumi_eta.eps}
\caption{The muon trigger efficiency for each trigger level (left) and fractional efficiency difference (right) as a function of $\eta$
in the \textit{low luminosity}  scenario using the MS-Probe.
The efficiencies determined with the Tag and Probe method are compared to those calculated in a Monte Carlo truth-based analysis.}
\label{TaP_Muon_Trigger_LowLumi_Eff_Diff_MSProbe}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%
\begin{table}[t]
\footnotesize
\centering
\begin{tabular}{|cccc|}
\hline
\textbf{Detector region} & \textbf{Barrel} & \textbf{Endcap} & \textbf{Overall} \\
& ($|\eta<1.05|$) & ($1.05 < |\eta| < 2.4$)    &  ($0 < |\eta| < 2.4$) \\
\hline
\hline
\multicolumn{4}{|c|}{\textbf{\textbf{Low luminosity - ID probe} \boldmath{($\int\mathcal{L}dt = 50~pb^{-1}$)}}} \\
\hline
Trigger Efficiency & 71.65 & 83.59 & 77.38 \\
\hline
\hline
Statistical Uncertainty & 0.42 & 0.36 & 0.28 \\
\hline
$|\epsilon_{TRUTH} - \epsilon_{TP}|$ & 0.23 & 0.40 & 0.10 \\
\hline
Expected Background Contribution & 0.57 & 0.17 & 0.40 \\
\hline
Overall Systematic Uncertainty & 0.61 & 0.43 & 0.41 \\
\hline
\multicolumn{4}{|c|}{\textbf{\textbf{Low luminosity - MS probe} \boldmath{($\int\mathcal{L}dt = 50~pb^{-1}$)}}} \\
\hline
\hline
Trigger Efficiency & 76.94 & 87.83 & 82.13 \\
\hline
Statistical Uncertainty & 0.41 & 0.34 & 0.27 \\
\hline
$|\epsilon_{TRUTH} - \epsilon_{TP}|$ & 0.17 & 0.64 & 0.33 \\
\hline
Expected Background Contribution & 0.01 & 0.00 & 0.01 \\
\hline
Overall Systematic Uncertainty & 0.17 & 0.64 & 0.33 \\
\hline
\end{tabular}
\caption{Estimated uncertainties of in-situ determined muon overall trigger efficiency for the \textit{low luminosity} scenario, using an ID- and an MS-Probe track. Systematic uncertainties are reported for background contribution and absolute difference with Monte Carlo truth-based analysis.}
\label{TaP_Muon_Trigger_LowLumi_OverAll_Table}
\end{table}
%%%%%%%%%%%%%%%%

\subsubsection{Low luminosity measurements}

The relative efficiency as a function of $\eta$,  measured at each trigger level,
%referred to the selection done by previous one,
is shown in Fig.~\ref{TaP_Muon_Trigger_LowLumi_Eff_IDProbe} using
the ID-probe. L1 acceptance losses are related to an incomplete
coverage of the trigger detectors due to the presence of support and
access structures. The L2 efficiency, with respect to the L1
selection, is about 96\% in the barrel region with a small decrease
in the endcap, an improvement is expected due to optimization  of
the TGC cabling in next software releases.
% (this is expected to improve
%by using different analysis techniques).
The EF shows an $\eta$ efficiency distribution, with respect  to L2,
close to 99\% in the barrel region and a very small decrease from
$|\eta| > 2.0$. In the region $1.05<|\eta|<1.3$ the absence of the
some MDT chambers in the ATLAS initial layout resulted in an
efficiency loss of about 10\%.\footnote{
The missing chambers are scheduled to be installed by the end of 2009.}
\\
The observed agreement with the Monte Carlo truth-based analysis is very good. In order to quantitatively estimate the bin-by-bin differences the fractional efficiency difference''
\begin{eqnarray}
\frac{\epsilon_{Tag\&Probe} - \epsilon_{MC}}{\epsilon_{MC}}
\label{TP_fed}
\end{eqnarray}
has been computed. This quantity is shown for each trigger level in Fig.~\ref{TaP_Muon_Trigger_LowLumi_Eff_IDProbe} as a function of $\eta$.\\
The agreement between Tag and Probe method and Monte Carlo analysis
is very high, more than 99\% over all the trigger coverage. The only
observed deviations, at the level of 2\%, are found in the central
crack at $\eta = 0$ and in the transitions from barrel to endcap at
$|\eta| = 1.05$. The results obtained by the application of the Tag
and Probe method using the standalone MS-Probe are reported in
Fig.~\ref{TaP_Muon_Trigger_LowLumi_Eff_Diff_MSProbe} as a functions
of $\eta$.
%Similar results have been obtained for the $\phi$ distributions.\\
With respect to the values shown in Fig.~\ref{TaP_Muon_Trigger_LowLumi_Eff_IDProbe} inefficiencies due to L1 acceptance cracks are partially factorized in the offline muon reconstruction efficiency of the MS-Probe (e.g. the $\eta = 0$ region.). %The factorization is not complete due to the better coverage of the precision chambers.
The same effect is clearly evident at the EF level for the
efficiency loss at $1.05<|\eta|<1.3$ visible in
Fig.~\ref{TaP_Muon_Trigger_LowLumi_Eff_IDProbe}.

Table \ref{TaP_Muon_Trigger_LowLumi_OverAll_Table} shows the
uncertainties on the overall trigger efficiency in each case,
calculated also only in barrel and endcap regions. The statistical
uncertainty is reported together with expected systematic errors.
Two sources of systematic uncertainties are considered: the absolute
difference with respect to the value measured in a Monte Carlo
truth-based analysis and the background contribution, evaluated by
comparing the efficiency calculated with Tag and Probe method using
only the signal sample and using a cross-section weighted sum of all
processes. Both systematics are less than 0.5\%. A greater
background contribution is observed when using  ID-Probe,
since the isolation is based only on Inner Detector quantities.

\subsubsection{High luminosity measurements}

%%%%%%%%%%%%%%%%%
%\begin{figure}[b!]
%\begin{center}
%\includegraphics[width=0.4\textwidth]{FiguresBellomo/Zmumu_MuonTrigger_Efficiency_mu20_3levels_IDProbe_thr2_HighLumi_eta.eps}
%\includegraphics[width=0.4\textwidth]{FiguresBellomo/Zmumu_MuonTrigger_EfficiencyDiff_mu20_3levels_IDProbe_thr2_HighLumi_eta.eps}
%\caption{Comparison of the muon trigger efficiency for each trigger level as a function of $\eta$ determined by the Tag and Probe method and by the Monte Carlo truth-based analysis in the \textit{high luminosity}  scenario using the ID-Probe (left) and their fractional difference (right).}
%\label{TaP_Muon_Trigger_HighLumi_Eff_Eta}
%\end{center}
%\end{figure}
After early data is collected and analyzed, a better understanding of the detector, in terms of calibration and alignment,
will allow to use all the available information such as Calorimeter quantities for track isolation and combination of Inner Detector and Muon Spectrometer tracks.
The trigger efficiency measurements from data in this scenario are reported using the \textit{high luminosity} dataset of
$\int\mathcal{L}dt = 1000~pb^{-1}$.
%The results obtained with the ID-Probe are illustrated in Fig.~\ref{TaP_Muon_Trigger_HighLumi_Eff_Eta} as a function of $\eta$.
%Similar values have been found also for the $\phi$ dependence and using the MS-Probe.

As in the \textit{low luminosity} case the measured differences
between the Tag and Probe and Monte Carlo analysis are compatible
with zero. Small deviations at the level of 1 to 2\% are observed in
the endcap for the L2 trigger efficiency in the $|\eta| > 1.5$ region. These effects are expected to be reduced by the  L2
algorithm optimization. Results are shown in Table
\ref{TaP_Muon_Trigger_HighLumi_OverAll_Table_IDProbe}. With the
addition of the calorimeter-based isolation, the background
systematic contribution is reduced by a factor of 10 with respect to
the low luminosity scenario.
%%%%%%%%%%%%%%%%%%
\begin{table}[t]
\footnotesize
\centering
\begin{tabular}{|cccc|}
\hline
\textbf{Detector region} & \textbf{Barrel} & \textbf{Endcap} & \textbf{Overall} \\
& ($|\eta<1.05|$) & ($1.05 < |\eta| < 2.4$)    &  ($0 < |\eta| < 2.4$) \\
\hline
\multicolumn{4}{|c|}{\textbf{\textbf{High luminosity - ID probe} \boldmath{($\int\mathcal{L}dt = 1000~pb^{-1}$)}}} \\
\hline
\hline
Trigger Efficiency & 73.24 & 86.31 & 79.73 \\
\hline
Statistical Uncertainty  & 0.10 & 0.08 & 0.06 \\
\hline
$|\epsilon_{TRUTH} - \epsilon_{TP}|$ & 0.02 & 0.72 & 0.58 \\
\hline
Expected Background Contribution & 0.05 & 0.01 & 0.03 \\
\hline
Overall Systematic Uncertainty & 0.05 & 0.72 & 0.58 \\
\hline
\end{tabular}
\caption{Estimated uncertainties of in-situ determined muon overall trigger efficiency for the \textit{high luminosity} scenario using the ID-Probe.
Systematic uncertainties are reported for background contribution and absolute difference with Monte Carlo truth-based analysis.}
\label{TaP_Muon_Trigger_HighLumi_OverAll_Table_IDProbe}
\end{table}
%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.49\textwidth]{FiguresBellomo/Zmumu_MuonTrigger_Efficiency_TurnOn_mu20_IDProbe_flat_HighLumi.eps}
\includegraphics[width=0.49\textwidth]{FiguresBellomo/Zmumu_MuonTrigger_EfficiencyDiff_mu20_3levels_IDProbe_flat_HighLumi_pt.eps}
\caption{Muon trigger efficiency turn-on curve after each trigger level determined by the Tag and Probe method
and by the Monte Carlo truth-based analysis in the \textit{high luminosity}  scenario using the ID-Probe (left).
In the right plot the fractional efficiency difference is shown.}
\label{TaP_Muon_Trigger_HighLumi_Eff_TurnOn_IDProbe}
\end{center}
\end{figure}
The dependence of the trigger efficiency on the $\pt$  shows the typical shape of a turn-on curve.
%, where the raising point is located
%near the given threshold (20~GeV for the item under analysis).
The sharpness of the curve is related to the finite $\pt$
resolution, $\sim$ 30\% at L1, $\sim$ 5\% at L2, and $\sim$ 3\% at
the EF.
%Turn-on curves have been measured from data with the \emph{high luminosity} sample, to have a better determination of the raising region.
Turn-on curves are shown in
Fig.~\ref{TaP_Muon_Trigger_HighLumi_Eff_TurnOn_IDProbe} using the
ID-probe and similar results are obtained with the MS-probe. The
turn-on point and the plateau values are correctly reproduced from
data. The fractional efficiency difference is shown for each trigger
level. The disagreement near the threshold is within 5\%, due mainly
to resolution effects, while in the plateau region the observed
difference is less than 1\%.

\section{High p$_{T}$ Dimuon final states}
\label{sec:highmass}

In principle, the high mass dilepton/diphoton
resonance search should have a  fairly straightforward trigger strategy  as there are very high energy leptons in the event. However,
there are several questions that remain: what trigger requirements are optimal for the analysis?
What $\pt$ thresholds and object quality selection should be applied?
How can one estimate the trigger efficiency from data for such rare (or non-existent) events?
Are the same object quality requirements that are appropriate for lower $p_{T}$ objects appropriate for very high energy objects?
%\begin{itemize}
%\item What trigger requirements are optimal for the analysis? What $p_{T}$ thresholds and object quality selection should be applied?
%\item How can one estimate the trigger efficiency from data for such rare (or non-existent) events?
%\item Are the same object quality requirements that are appropriate from 'lower' $p_{T}$ objects appropriate for very high energy objects?
%\end{itemize}

This Section addresses these questions, evaluates the trigger
efficiency for the signal samples of interest, and discusses the
trigger strategy for the earliest data taking  periods. It is
expected that during both low and high luminosity periods there will
be an unprescaled single muon trigger without an isolation
requirement. The 20 or 40 GeV  threshold are expected to be highly
efficient for a high mass resonance decaying into two muons.

\subsection{Efficiency estimate}

The muon trigger efficiency is estimated using several  methods.
The first method is to rely on  simulation;
%as part of the standard
%ATLAS reconstruction software from a full GEANT detector simulation and an emulation/simulation of the ATLAS muon trigger. W
while this is the simplest and
most direct method it
%relies completely on simulation techniques which are
is believed to be somewhat more optimistic (better resolution, higher efficiency).
%than results from real data from previous experiments at the Tevatron and before.
Therefore, the trigger efficiency  is also estimated  using methods
which can be applied to real data.

The trigger efficiencies are calculated with respect to the offline
event selection. Two combined muons  are required to satisfy the
cuts:  $|\eta|~<~2.7$,   $\pt~>~30$ GeV, track fit $\frac{ \chi^{2}}{\rm{D.O.F}}~< 10$ and Inner Detector and Muon Spectrometer
track match $\frac{\chi^{2}}{\rm{D.O.F}}~<10$.
%\begin{itemize}
%\item $|\eta|~<~2.7$
%\item $\pt~>~30$ GeV
%\item Track fit $\frac{ \chi^{2}}{\rm{D.O.F}}~< 10$
%\item Inner Detector and Muon Spectrometer track match $\frac{\chi^{2}}{\rm{D.O.F}}~<10$
%\end{itemize}
The trigger efficiencies for the dimuon heavy resonance Monte Carlo samples are shown in Table \ref{table:trig_eff}.
%The trigger efficiency quoted is simply the fraction of events satisfying the offline requirements which also fired
%the corresponding trigger.
\begin{table}
\centering
\begin{tabular}{|l|l|c|c|c|c|} \hline
%\multicolumn{6}{c|} {mu20 Efficiency } \\ \hline
Sample & L1 \% & L2  \% & EF  \% & Total Trigger Efficiency \% \\ \hline
400 GeV $\rho_{T}/\omega_{T}$ & 97.6 $\pm$ 0.10 & 98.8 $\pm$ 0.07 & 99.5 $\pm$ 0.05 & 96.0 $\pm$ 0.13   \\
600 GeV $\rho_{T}/\omega_{T}$ & 98.1 $\pm$ 0.08 & 98.5 $\pm$ 0.08 & 99.2 $\pm$ 0.06 & 95.9 $\pm$ 0.13   \\
800 GeV $\rho_{T}/\omega_{T}$ & 97.6 $\pm$ 0.10 & 98.7 $\pm$ 0.07 & 99.2 $\pm$ 0.05 & 95.6 $\pm$ 0.13  \\
1 TeV $\rho_{T}/\omega_{T}$   & 97.6 $\pm$ 0.09 & 98.7 $\pm$ 0.07 & 99.2 $\pm$ 0.05 & 95.6 $\pm$ 0.12  \\
%1 TeV Z' (SSM)     & ?? & ?? & ?? & ??  \\
1 TeV Z' (E6)      & 97.8 $\pm$ 0.09 & 98.9 $\pm$ 0.06 & 99.5 $\pm$ 0.04 & 96.3 $\pm$ 0.1  \\
2 TeV Z' (SSM)     & 97.6 $\pm$ 0.14 & 98.7 $\pm$ 0.11 & 98.9 $\pm$ 0.10 & 95.3 $\pm$ 0.2  \\ \hline\hline
\end{tabular}
\caption{Trigger efficiencies of dimuon resonance samples. For the
meaning of E6 and SSM see \protect  \cite{DiLepNote}.
\label{table:trig_eff} }
\end{table}
%Figure ~\ref{fig:Zprime_L1TrigEff} shows the L1 trigger efficiency with respect to offline reconstruction as a function of $\phi$, $\eta$, and $p_{T}$.
%There are several features of the efficiency distributions. First, notice the that the efficiency is lower in the barrel region than in the end-cap. This
%is due to the incomplete trigger coverage in the barrel with respect to the precision chambers. The presence of support structures and services lead to
%incomplete geometric acceptance of the detector. As discussed in ~\cite{muon}, the algorithmic efficiency is about 99\% for muons within the trigger
%acceptance. Secondly note the two dips in the efficiency as a function of phi. This is caused by the presence of the 'feet' support structure which
%leads to incomplete trigger coverage. Finally, note that the trigger chambers  cover the region of $|\eta|~<~2.4$ while the CSC (cathode strip
%chambers) allow offline muon reconstruction up to $\eta$ of 2.7.
The efficiency as a function of $\pt$ has been fit to
% function parameterizing the efficiency as a function of transverse momentum:
%Since
%the estimated muon momenta has a finite resolution one expects the estimate to be Gaussian distributed about the true muon momentum. Thus by
%selecting the muons above a given threshold one is integrating the Gaussian distribution above some cut-off. The parameterization is written as:
\begin{equation}
f(p_{T}) = 0.5 \cdot A_{2} \cdot (1.0 + erf(\frac{p_{T} - A_{0}}{\sqrt{2} \cdot A_{1}}))
\label{eqn:pt_eff}
\end{equation}
where $erf$ is the error function, $A_{0}$, $A_{1}$, and $A_{2}$ are the fit parameters which represent the $p_{T}$ value at which
the efficiency reaches half its maximum value, the slope of the turn-on curve, and the maximum efficiency in the plateau region,
respectively.
%The statistical
%uncertainty on the trigger efficiency as a function of $p_{T}$ can be written as :
%
%
%\Delta f^{2} = (\frac{\delta f}{\delta A_{0}})^{2} \cdot (\Delta A_{0})^{2} + \cdot  (\frac{\delta f}{\delta A_{1}})^{2} \cdot (\Delta A_{1})^{2}
%+ (\frac{\delta f}{\delta A_{2}})^{2} \cdot (\Delta A_{2})^{2}
%
%
%which is can be written:
%
%
%\Delta f^{2} = [\frac{A_{2}}{\sqrt{2 \pi} A_{1}} \cdot e^{- (\frac{ p_{T} - A_{0} }{ \sqrt{2} \cdot A_{1} })^{2}}]^{2} (\Delta A_{0})^{2} \\
%+ [\frac{A_{2} \cdot (p_{T} - A_{0})}{\sqrt{2\pi} A_{1}^{2}} \cdot e^{- (\frac{p_{T} - A_{0}}{\sqrt{2} \cdot A_{1}})^{2}}]^{2}  (\Delta A_{1})^{2} \\
%+ [ 0.5 \cdot (1.0 + erf(\frac{p_{T} - A_{0}}{\sqrt{2} \cdot A_{1}}))]^{2} (\Delta A_{2})^{2}
%
%
%where $\Delta A_{0}$, $\Delta A_{1}$, and $\Delta A_{2}$ are the statistical uncertainties on the fit parameters.

%\begin{figure}
%\begin{center}
%\begin{tabular}{ccc}
%\includegraphics[width=5.5cm]{plots/L1PhiEff_Zprime.eps} &
%\includegraphics[width=5.5cm]{plots/L1EtaEff_Zprime.eps} &
%\includegraphics[width=5.5cm]{plots/L1PtEff_Zprime.eps}
%\end{tabular}
%\end{center}
%\caption {L1 Trigger Efficiency as a function of $\phi$ (right), $\eta$ (center), and $p_{T}$ with respect to
%offline event selection for the 1 TeV SSM Z' sample \label{fig:Zprime_L1TrigEff}.}
%\end{figure}
%
%\begin{figure}
%\begin{center}
%\begin{tabular}{ccc}
%\includegraphics[width=5.5cm]{plots/L2PhiEff_Zprime.eps} &
%\includegraphics[width=5.5cm]{plots/L2EtaEff_Zprime.eps} &
%\includegraphics[width=5.5cm]{plots/L2PtEff_Zprime.eps}
%\end{tabular}
%\end{center}
%\caption {L2 Trigger Efficiency as a function of $\phi$ (right), $\eta$ (center), and $p_{T}$ with respect to
%offline event selection for the 1 TeV SSM Z' sample \label{fig:Zprime_L2TrigEff}.}
%\end{figure}
%
%\begin{figure}
%\begin{center}
%\begin{tabular}{ccc}
%\includegraphics[width=5.5cm]{plots/EFPhiEff_Zprime.eps} &
%\includegraphics[width=5.5cm]{plots/EFEtaEff_Zprime.eps} &
%\includegraphics[width=5.5cm]{plots/EFPtEff_Zprime.eps}
%\end{tabular}
%\end{center}
%\caption {EF Trigger Efficiency as a function of $\phi$ (right), $\eta$ (center), and $p_{T}$ with respect to
%offline event selection for the 1 TeV SSM Z' sample \label{fig:Zprime_EFTrigEff}.}
%\end{figure}
There are several  methods to evaluate the trigger efficiency from the data itself.
A possible one is to look at the trigger efficiency
for a known experimentally clean signature that is similar to the final state of interest;
\Zmumu
%$Z \rightarrow \mu \mu$
is one of such signatures.
%In this case the only natural choice is to examine the
%trigger efficiency of the $Z \rightarrow \mu \mu$ final state. Here one can obtain a very clean signature which is very similar to the hypothetical
%heavy resonance (the difference being the mass of the resonance).
Since the Z  is light compared to the total center of mass energy, it can
be produced with a significant $p_{T}$ distribution.
The trigger efficiency on the Z  can be measured and
extrapolated to high $\pt$.
The advantage of this method is that it uses data  to measure the trigger efficiency which is the
most accurate method of measuring the Z  trigger efficiency.
%However, the method does come with caveats.
A disadvantage is that the muon trigger efficiency
is being extrapolated to a $\pt$ by a factor of 10 higher  than the mean $\pt$ of the muons from the Z  decay.
%It is hoped that by checking the efficiency in several different methods the most accurate estimate can be achieved.

The strategy of evaluating the trigger efficiency from data is as
follows. It is first necessary to use one of several methods to
estimate the muon trigger efficiency as a function of the muon
$p_{T}$ and its uncertainty. The single object trigger efficiency
allow the construction of the probability for an event with N
objects to pass the trigger. This probability can be written as:
\begin{equation}
P = 1 - \prod_{i=1}^{N} (1 - P_{i})
\label{eqn:event_prob}
\end{equation}
\noindent
where $P_{i}$ is the probability for the $i$-th object to pass the trigger.

Two common methods that have been used extensively at the Tevatron
are the selection by orthogonal triggers and the 'Tag and Probe'
method using Z $\rightarrow~\mu\mu$ decay.
%\begin{figure}
%\begin{center}
%\begin{tabular}{ccc}
%\includegraphics[width=5.5cm]{plots/L1PhiEff_Zmumu.eps} &
%\includegraphics[width=5.5cm]{plots/L1EtaEff_Zmumu.eps} &
%\includegraphics[width=5.5cm]{plots/L1PtEff_Zmumu.eps}
%\end{tabular}
%\end{center}
%\caption {L1 Trigger Efficiency as a function of $\phi$ (right), $\eta$ (center), and $p_{T}$ with respect to
%offline event selection \label{fig:Zmumu_L1TrigEff}.}
%\end{figure}
%
%
%\begin{figure}
%\begin{center}
%\begin{tabular}{ccc}
%\includegraphics[width=5.5cm]{plots/L2PhiEff_Zmumu.eps} &
%\includegraphics[width=5.5cm]{plots/L2EtaEff_Zmumu.eps} &
%\includegraphics[width=5.5cm]{plots/L2ptEff_Zmumu.eps}
%\end{tabular}
%\end{center}
%\caption {L2 Trigger Efficiency as a function of $\phi$ (right), $\eta$ (center), and $p_{T}$ with respect to
%offline event selection \label{fig:Zmumu_L2TrigEff}.}
%\end{figure}
%
%\begin{figure}
%\begin{center}
%\begin{tabular}{ccc}
%\includegraphics[width=5.5cm]{plots/EFPhiEff_Zmumu.eps} &
%\includegraphics[width=5.5cm]{plots/EFEtaEff_Zmumu.eps} &
%\includegraphics[width=5.5cm]{plots/EFPtEff_Zmumu.eps}
%\end{tabular}
%\end{center}
%\caption {EF Trigger Efficiency as a function of $\phi$ (right), $\eta$ (center), and $p_{T}$ with respect to
%offline event selection \label{fig:Zmumu_EFTrigEff}.}
%\end{figure}
%The 'Tag and Probe' method is a simple way of using the known Z  resonance as a 'candle' to measure high $p_{T}$ lepton
%properties.
%The method requires two offline muons to have an invariant mass within several sigma of the known Z  mass.
The 'Tag and Probe' requires two offline muons to have an invariant
mass within 12 GeV$^2$ of 91.1 GeV$^2$.
%For the event is recorded in the first place
%at least one of the muons must satisfy the trigger requirement. In order to avoid any biases, one muon is randomly assigned
%as the tag muon' while the other becomes the 'probe'. It is then checked if the probe muon has passed the muon trigger.
%The efficiency obtained from the tag and probe method for  mu20 is shown in Figures ~\ref{fig:Zmumu_L1TrigEff}, ~\ref{fig:Zmumu_L2TrigEff}
%and ~\ref{fig:Zmumu_EFTrigEff}.
The turn on curves as a function of the offline muon \pt, obtained
using this method, is fit  to equation ~\ref{eqn:pt_eff}. This
procedure was repeated for all three trigger levels and the results
are summarized in Table~\ref{tab:trigEffFit_tab}.
%Note that all of the efficiency plots show above are not
%event efficiences but rather for one of the muons from the Z. The probability that the event pass the trigger is significantly higher
%because the single muon trigger requires at least one muon to pass the trigger while each event has two offline muons. The L1 trigger
%efficiency is quoted with respect to the offline reconstruction. The L2 trigger efficiency is quoted with respect to the events which passed
%the L1 trigger and the EF with respect to events that passed both the L1 and L2 trigger.

\begin{table}
\centering
\begin{tabular}{|c|c|c|c|} \hline \hline
Trigger Level & $A_{0}$ & $A_{1}$ & $A_{2}$ \\ \hline\hline
L1          & 12.5 $\pm$ 0.3 & 3.7  $\pm$ 0.4 & 0.845 $\pm$ 0.02 \\ \hline
L2          & 19.6 $\pm$ 0.2 & 1.59 $\pm$ 0.19 & 0.976 $\pm$ 0.02 \\ \hline
EF            & 19.5 $\pm$ 0.4 & 1.56 $\pm$ 0.3  & 0.931 $\pm$ 0.01 \\ \hline
\end{tabular}
\caption{ Fitted parameter for the L1, L2, and EF of the trigger $\pt$ turn on curves ~\label{tab:trigEffFit_tab}
}.
\end{table}

A second possible method of evaluating the trigger efficiency with
data is by the method of orthogonal triggers. To obtain a sample of
unbiased events we select events that pass one of the calorimeter
based triggers, the single 20 GeV jet trigger. We then perform the
offline analysis and require that we have a dimuon pair using
identical event selection to the 'Tag and Probe' analysis. From this
sample we simply check the fraction of events that pass the L1, L2,
and EF trigger conditions for the 20 GeV muon trigger.  The results
are shown in Table~\ref{table:mu_jetTrig} and are in good agreement
with the 'Tag and Probe' method and direct emulation of the trigger
on the Monte Carlo sample. Unfortunately, in the real experiment a
single jet trigger with a threshold of 20 GeV would be very highly
prescaled and hence will suffer from poor statistics. Events that
passed any calorimeter trigger could be used for this study if
biases in the event topology were taken into account, however, such
a study is beyond the scope of this note.

We have developed two methods that could be used to evaluate the
trigger efficiency from data. Extraction of the muon trigger
efficiency as a function of the reconstructed muon kinematics via a
tag and probe method and an orthogonal trigger method agree well
with the simulated trigger efficiency. These methods will allow us
to more accurately estimate the trigger efficiency for LHC data.

\begin{table}
\centering
\begin{tabular}{|c|c|c|c|c|} \hline\hline
\emph{Sample} & L1Mu20 Efficiency \% & L2Mu20 Efficiency \% & EFMu20 Efficiency & Total Efficiency \\ \hline\hline
Z' 1 TeV (SSM) & 97.7 $\pm$ 0.11 & 99.0 $\pm$ 0.07 & 99.6 $\pm$ 0.04 & 96.3 $\pm$ 0.01 \\ \hline
\Zmumu
%Z $\rightarrow \mu \mu$
& 97.83 $\pm$ 0.04  & 98.86 $\pm$ 0.03 & 99.52 $\pm$ 0.02 & 96.26 $\pm$ 0.05\\ \hline \hline
\end{tabular}
\caption{ L1Mu20 trigger efficiencies at L1, L2, and Event Filter  w.r.t offline reconstruction
using orthogonal trigger selection to record events ~\label{table:mu_jetTrig}}.
\end{table}

%\end{document}

%%%%\input{DA}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Summary and conclusion
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

\section{Summary}

In this paper Muon trigger baseline performance and rates for initial and standard LHC operation
have been presented.
Trigger efficiency has been studied in detail in a wide energy range using single muon simulated samples.
From efficiencies the muon rates have been evaluated.
It should be noted that due to the uncertainties of the inclusive muons cross-sections,
rates could vary significantly and different threshold cuts could be adopted.
A further rate reduction should come from dedicated strategies to reject muon from in-flight decays of $K$ and $\pi$;
in this paper a preliminary analysis is presented at Event Filter.
It is demonstrated that a good rejection can be achieved with contained losses of prompt muons.

The possibility to select at the ATLAS second level trigger with high efficiency isolated muons from $W$ and $Z$ decays
reducing the ones from heavy quark decays has been studied in depth. Although electronic readout
and pileup noise have been simulated, no cavern background has been yet included.
A factor of ten reduction on high-\pt\ muons from heavy-quark decays has been obtained
while maintaining a $95\%$ efficiency
on $Z\to\mu^+\mu^-$ final state.
Next step will be to investigate how much the use of the
longitudinal granularity of the calorimeters and inner tracker detector will increase the muon isolation rejection power.

The overall performance of the TileCal muon tagging algorithm has been presented,
using MC samples of single muons and inclusive B-Physics processes,    including  minimum-bias pileup at low luminosity.

We finally addressed the question of how the muon trigger efficiency
can be measured with \Zmumu
%$Z \rightarrow \mu\mu$
and $Z^\prime \rightarrow \mu\mu$ using the tag and probe method. This technique shows
a very good agreement with results based on Monte Carlo studies.

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Acknowledgements
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\section{Acknowledgements}

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Bibliography
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Style file to use with mcite.
% Use atlasstyle with just cite.
\bibliographystyle{atlasstylem}
\begin{thebibliography} {1}

%\bibitem{Ganga} https://twiki.cern.ch/twiki/\-bin/\-view/Atlas/\-Distributed\-Analysis\-Using\-Ganga
%\bibitem{Panda} https://twiki.cern.ch/twiki/\-bin/view/\-Atlas/PanDA

\bibitem{muon} ATLAS Collaboration,
\newblock Muon Spectrometer Technical Design Report, CERN/LHCC/97-022 (1997).

\bibitem{pisa_meet} A.~Sidoti,
\newblock The ATLAS trigger muon 'vertical slice, Nucl.\ Instrum.\ Meth.\  A {\bf 572} (2007) 139.
%%CITATION = NUIMA,A572,139;%%

\bibitem{pythia6} T.~Sjostrand, S.~Mrenna and P.~Skands,
\newblock  PYTHIA 6.4 physics and manual, JHEP {\bf 0605} (2006) 026.
%  [arXiv:hep-ph/0603175].
%%CITATION =JHEPA,0605,026;%%

\bibitem{ATLASHLTTDR} ATLAS Collaboration,
\newblock ATLAS High-Level Trigger, Data Acquisition and Controls Technical Design Report, CERN/LHCC/2003-022 (2003).

\bibitem{pythia5}
T.~Sjostrand,
\newblock Pythia 5.7 And Jetset 7.4: Physics And Manual,  CERN-TH-7112-93-REV (1995).
%  arXiv:hep-ph/9508391
%%CITATION = HEP-PH/9508391;%%

\bibitem{dpmjet} J.Ranft,
\newblock DPMJET version H3 and H4, INFN-AE-97-45 (1997).

\bibitem{BphyKPI} ATLAS Collaboration,
\newblock Triggering on low-\pt\ muons and dimuons for B Physics, CSC Note.

\bibitem{muiso_mvb} BABAR Collaboration (P.F. Harrison and H. Quinn (editors) et al.),
\newblock The BABAR Physics Book, SLAC-R-0504 (1998).

\bibitem{TileMuId} G. Usai
\newblock  Nucl. Instrum. Meth. 518 (2004) 36.

\bibitem{AtlasLVL1TDR} ATLAS Collaboration,
\newblock First Level Trigger Technical Design Report, CERN/LHCC/98-14 (1998).

\bibitem{WZXSecCSCNote} ATLAS Collaboration,
\newblock Electroweak Boson Cross-Section Measurements, this volume.

\bibitem{DiLepNote} ATLAS Collaboration,
\newblock DiLepton Resonances at High Masses, this volume.

%\bibitem{ATLAS_physics}{\it ATLAS Detector and Physics Performance Techinical Design Report} CERN/LHCC99-14/15
%\bibitem{b_physics}{\it ATLAS : B Physics Reach}  Eur. Phys. J., C {\bf 34} (2004) s385-s392
%\bibitem{background}{\it Benchmarking the Particle Background in the LHC Experiments} [CERN-THESIS-2002-001].
%\bibitem{CMA} {\it The Coincidence Matrix ASIC of the L1 Muon
%Barrel Trigger of the ATLAS Experiment} IEEE Transactions on Nuclear Science, August 2003 Issue vol. 50, no. 4

%% trigmoore description
%\bibitem{moore} D. Adams et al., \emph{Track Reconstruction in the ATLAS Muon %%@
%Spectrometer with MOORE}, ATLAS Note, ATL-SOFT-2003-007, 2003.
%\bibitem{muid} Th. Lagouri et al, {\it A Muon Identification and Combined Reconstruction Procedure for the ATLAS detector at CERN LHC''}, IEEE Trans.Nucl.Sci., 51 (2004) 3030-3033.
%\bibitem{RegSel}
%V. Boisvert et al., {\it A New Implementation of the Region-of-
%Interest Strategy for the ATLAS Second Level Trigger''}, ATLAS
%Note ATL-DAQ-2003-034.
%\bibitem{tmoore} D. Adams et al., {\it MOORE as Event Filter in the ATLAS High Level Trigger''}, ATLAS Note,
%ATL-SOFT-2003-008, 2003,

%\bibitem{tmoore1} G.Cataldi et al.{\it Muon identification with the event filter of the ATLAS experiment at CERN LHC''},
%IEEE Trans.Nucl.Sci.53:870-875,2006.
%\bibitem{HLT} M. Elsing et al,
%{\it Analysis and Conceptual Design of the HLT Selection Software''}, ATLAS Note,ATL-DAQ-2002-013,2002.

%% slice conf

%\bibitem{sitrack} M. Cervetto et al., {\it SiTrack: a LVL2 track reconstruction algorithm based on Silicon detectors''},
% ATLAS Communication ATL-COM-DAQ-2003-025

%\bibitem{idscan} N. P. Konstantinidis, {\it A Fast Tracking Algorithm for the ATLAS Level 2 Trigger''},
%Nucl. Instrum. Methods Phys. Res., A 566 (2006) 166-169.

%\bibitem{ipat} https://twiki.cern.ch/twiki/bin/view/Atlas/IPatRec

%\bibitem{EFID} T. Cornelissen et al., {\it Concepts, Design and Implementation of the ATLAS New Tracking''},
%ATL-SOFT-PUB-2007-007 (2007).

%%rates

%\bibitem{bcxs} A. Dewhurst et al.,{\it Low \pt\ muon and dimuon rates in ATLAS''}, ATLAS Comminication ATL-COM-PHYS-2007-089

%\bibitem{ROD} J. Castelo et al., {\it TileCal ROD Hardware and Software Requirements''}, ATLAS Note ATL-TILECAL-2005-003.

%\bibitem{IDScan-1} H. Drevermann and N. Konstantinidis, {\it Determination of the z position of priminary interactions in ATLAS''}, ATLAS Note ATL-DAQ-2002-014 (2002).

%\bibitem{IDScan-2} H. Drevermann and N. Konstantinidis, {\it Algorithms to select space points of tracks from single primary interactions in ATLAS''}, ATLAS Note ATL-COM-DAQ-2003-040 (2003).

%\bibitem{eerola} Eerola, Paule Anna Mari, {\it The inclusive muon cross-section in ATLAS''},
%    ATLAS Note ATL-MUON-98-222; ATL-M-PN-222 (1998)

\end{thebibliography}

\end{document}
%
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% Author List
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\newpage
%\input /atlas/paper/authorlist.tex
\newpage

%
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% Technical Aspects
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

%\begin{center}
%%Subfigure
%\subfigure[Radial Weights from FCal 1 module]{
%\label{F:pion_rw1}
%\includegraphics[width=
%0.55\textwidth]{atlas_subfigure1.eps}
%}
%%Subfigure
%\subfigure[Radial Weights from FCal 2 module]{
%\label{F:pion_rw2}
%\includegraphics[width=
%0.55\textwidth]{atlas_subfigure2.eps}
%}
%%Subfigure
%\subfigure[Radial Weights from FCal 3 module]{
%\label{F:pion_rw3}
%\includegraphics[width=
%0.55\textwidth]{atlas_subfigure3.eps}
%}
%\caption{Radial weights for FCal 1 (\ref{F:pion_rw1}), FCal 2
%(\ref{F:pion_rw2}), and FCal
%3 (\ref{F:pion_rw3}).
%\label{fig:subfigexample}}
%\end{center}
%\end{figure}



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