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\begin{document}

\section*{Abstract}

\section{Introduction}
% Motiver l'utilisation d'un simulateur rapide
% - 1) rapide VS lent
% - 2) relativement bonne prédiction en première approximation
% - 3) permet de comparer
Full simulation of the response of large detectors components to high energy particles requires a lot of computing resources. Moreover, a good knowledge of the exact geometry of subdetectors and dead material content is mandatory. 

Fast simulation can be a  powerful predictive tool for typical response of a large detector in high energy collider.

The fast simulation of the detector response takes into account geometrical
acceptance of sub-detectors and their finite energy resolution, no smearing is
applied on particle direction. Charged particles, once are in the fiducial
volume of the detector are assumed to be reconstructed with $100\%$ probability.
The energy of each particle produced after hadronization, with a lifetime
$c\tau$ bigger than $10~\textrm{mm}$ is then smeared according to detectors along
particule's direction. For particles with a short lifetime such as the $K_s$,
the fraction of electromagnetic or hadronic energy is determined according to
its decay products. The calorimeter is assumed to cover the pseudorapidity range
$|\eta|<3$ and consists in an electromagnetic and an hadronic part. The energy
resolution is given by $\sigma_{E}/E=0.05/\sqrt{E} \oplus 0.25/E \oplus 0.0055$
for the electromagnetic part and by $\sigma_{E}/E=0.91/\sqrt{E}\oplus 0.038$ for
the hadronic part, where the energy is given in GeV. A very forward calorimeter
is assumed to cover $3<|\eta|<5$ with an electromagnetic and hadronic energy
resolution function given by $\sigma_{E}/E=1.5/\sqrt{E}\oplus 0.06$ and
$\sigma_{E}/E=2.7/\sqrt{E}\oplus 0.13$ respectively.\\

\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.7\textwidth]{detectorAng.eps}
\caption{\small{detectorAng.eps}}
\label{h_WW_ss_cut1}
\end{center}
\end{figure}


The acceptance cuts applied on leptons and jets used in this section are the
following :\\

\begin{itemize}

\item Electrons and muons are reconstructed if they fall into the acceptance of
the tracker, assumed to be $|\eta|<2.5$, and have to have a transverse momentum
above 10~GeV (the energy resolution of muons is taken to be the same as for
electrons). Lepton isolation demands that there is no other charged particles
with $p_T>2$~GeV within a cone of $\Delta R<0.5$ around the lepton.\\

\item Jets are reconstructed using a cone algorithm with $R=0.7$ and make only
use of the smeared particle momenta. The reconstructed jets are required to have
a transverse momentum above 20~GeV and $|\eta|<3.0$. A jet is tagged as $b$-jets
if its direction lies in the acceptance of the tracker, $|\eta|<0.5$, and if it
is associated to a parent $b$-quark. A $b$-tagging efficiency of $40\%$ is
assumed if the jet has a parent $b$ quark. For $c$-jets and light/gluon jets, a
fake
b-tagging efficiency of $10 \%$ and $1 \%$ respectively is assumed.\\

\item A jet is tagged as a $\tau$-jet if more than $90\%$ of its energy is
localized in a cone of $\Delta R=0.15$ around its axis. Moreover, this jet must
have its direction in the acceptance of the tracker and have exactly one charged
particle with $p_{T}>2$~GeV within a cone $\Delta R<0.4$ around the jet axis.
This procedure selects taus decaying hadronically with a typical efficiency of
$60\%$. Moreover, the minimal $p_T$ of the $\tau$-jet is required to be
10~GeV.\\

\end{itemize}

\section{implementation}
\subsection{Electron smearing}
The smearing of the electron 4-momentum $p^\mu$ is
 - if the electron is in the tracker ($\eta < MAX\_TRACKER$)
	Gaussian smearing with $\sigma = ELG_Ccen*E \oplus ELG_Ncen \oplus
ELG_Scen*\sqrt{E}$
 - else Gaussian smearing with $\sigma = ELS_Cfwd*E \oplus ELG_Sfwd*\sqrt{E}$
function \texttt{SmearElectron}
Only the energy $E$ is smeared, but neither $\eta$ nor $\phi$.
No negative values for the energy after smearing. If so, the $4$-momentum is set
to $(0,0,0,0)$.
\textbf{For the moment, electrons with $|\eta|> 5$ are also smeared !!!}

\subsection{Muon smearing}
The smearing ot the muon 4-momentum $p^\mu$ is given
by a Gaussian smearing of the $p_T$
function \texttt{SmearMuon}
Only the $p_T$ is smeared, but neither $\eta$ nor $\phi$.
No negative values for the energy after smearing. If so, the 4-momentum is set
to $(0,0,0,0)$.

\subsection{Hadron smearing}
The energy of the hadron is smeared in the following ways:
 - if the hadron is in the central calorimeter (eta < MAX\_CALO\_CEN)
	Gaussian smearing with $\sigma = HAD_Chcal*E_{hcal} \oplus HAD_Nhcal
\oplus HAD_Shcal*\sqrt{E_{hcal}} 
			               + HAD_Cecal*E_{ecal} \oplus HAD_Necal
\oplus HAD_Secal*\sqrt{E_{ecal}}$ 
	where $E_{hcal} + E_{ecal} = E$. As some long-living particles decay in
the calorimeters,
	some of them decay mostly in the ECAL, some mostly in the HCAL.
$E_{hcal}$ and $E_{ecal}$ are 
	given by $E_{hcal} = E \times F$ and $E_{ecal} = E times (1-F)$, where
$F$ is a fraction 
	$0 \leq F \leq 1$ describing each particles. By default, $F=1.$ but is
$F=0.7$ for $K^0_S$ and $\Lambda$.
 - if the hadron is somewhere else (\textbf{even outside the forward
calorimeters !!!})
	Gaussian smearing with $\sigma = HAD_Chf*E \oplus HAD_Nhf \oplus
HAD_Shf*\sqrt{E}$ 

Ainsi, pour les particules considérées comme stables par PYTHIA
mais non stables dans un détecteur tel CMS ($c\tau < 4m$), les
dépôts laissés dans les différents détecteurs sont directement
liés aux modes de désintégrations de ces particules. Les
hypothèses des dépôts d'énergie sont données dans le tableau
\ref{depot}.\newline

\begin{table}[!h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\emph{Particules stables} & \emph{Stable} &\emph{Mode de}
&$\Gamma_{i}/\Gamma$&\emph{Dépot}&\emph{Dépot}\\
\emph{dans PYTHIA}&\emph{dans
CMS}&\emph{desintégration}&&\emph{ECAL}&\emph{HCAL}\\\hline\hline
$\pi^{\pm}$ & oui & & & 0 & 1 \\ \hline $K^{\pm}$ & oui & & & 0 &
1 \\\hline $K^{0}_{S}$ & non & $\gamma\gamma\gamma\gamma$ & 0.31 &
0.3 & 0.7\\
& & $\pi^{+}\pi^{-}$& 0.69 &&\\\hline $K^{0}_{L}$ & oui & & & 0 &
1\\\hline $\Lambda^{0}$ & non & $\pi^{-}p/\pi^{+}\overline{p}$ &
0.64&
0.3 & 0.7\\
& & $n\pi^{0}$ & 0.36 & & \\\hline $\gamma$& oui & & & 1
&0\\\hline
\end{tabular}
\caption{Hypothèses des dépôts d'énergie pour les particules les
plus abondantes des jets.} \label{depot}
\end{center}
\end{table}


function \texttt{SmearHadron}
There is no ecal-hcal separation in the forward calorimeter.
No negative values for the energy after smearing. If so, the 4-momentum is set
to $(0,0,0,0)$.

\subsection{Calorimetric towers}
All final particles, which are neither muons nor neutrinos are produce a
calorimetric tower.
The same particles enter in the calculation of the missing transverse energy.
\textit{what is used is the particle smeared momentum, not the calorimetric
towers!}

\subsection{Tracks}
All final charged particles

\subsection{Time of flight}
Some subdetectors have the ability to measure the time of flight of the particle.
This correspond to the delay after which the particle is observed in the detector, after the bunch crossing.
The time of flight measurement of ZDC and FP420 detector is implemented here.
For the ZDC, the formula is simply
\begin{equation}
 t_2 = t_1 + \frac{1}{v} \times \big( \frac{s-z}{\cos \theta}\big),
\end{equation}
where $t_2$ is the time of flight, $t_1$ is the true time coordinate of the vertex from which the particle originates, $v$ the particle velocity, $s$ is the ZDC distance to the interaction point, $z$ is the longitudinal coordinate of the vertex from which the particle comes from, $theta$ is the particle emission angle. This assumes that the neutral particle observed in the ZDC is highly relativistic, i.e. travelling at the speed of light $c$. We also assume that $\cos \theta = 1$, i.e. $\theta \approx 0$ or equivalently $\eta$ is large. As an example, $\eta = 5$ leads to $\theta = 0.013$ and $1 - \cos \theta < 10^{-4}$.
The formula then reduces to
\begin{equation}
 t_2 = \frac{1}{c} \times (s-z)
\end{equation}
NB : for the moment, only neutrons and photons are assumed to be able to reach the ZDC. All other particles are neglected

To fix the ideas, if the ZDC is located at $s=140~\textrm{m}$, neglecting $z$ and $\theta$, and assuming that $v=c$, one gets $t=0.47~\mu\textrm{s}$.

\subsection{Tau identification}

Two ways to identify a tau : using the energy inside a cone or the number of
tracks in the cone.
\begin{itemize}
\item From the energy in the cone of radius TAU\_CONE\_ENERGY. To be taken into
account, a calo tower should (1) have a transverse energy $E_T = \sqrt{E_X^2 +
E_Y^2}$ above a given threshold M\_SEEDTHRESHOLD, (2) be inside a cone with a
radius R and the axis defined by (eta,phi).
\item From the number of tracks in the cone of radius TAU\_CONE\_TRACKS. To be
taken into account, a track should (1) have a transverse momentum $ p_T =
\sqrt{p_X^2 + p_Y^2} $ above a given threshold, (2) be inside a cone with a
radius R and the axis defined by (eta,phi).
\end{itemize}

\begin{wrapfigure}{l}{0.3\textwidth}
\includegraphics[width=0.3\textwidth]{Tau.eps}
\caption{\small{detectorAng.eps}}
\label{h_WW_ss_cut1}
\end{wrapfigure}


To identify a tau, one requires the \textit{electromagnetic collimation} and the
\textit{tracking isolation}.
The electromagnetic collimation is a kind of calorimetric isolation required
around the jet axis.
One requires that most of the energy of the cone is located in a small cone in
the middle of the jet cone:
 \begin{equation}
   %C_{\tau}^{e.m.} = \frac{ \Sum E_T^{cell}(\Delta R= TAU\_CONE\_ENERGY)}{ 
\sum E_T^{cell} (\Delta R= CONE\_RADIUS) > TAU\_EM\_COLLIMATION
 \end{equation}
 Typical values are TAU\_CONE\_ENERGY=0.15 ,  CONE\_RADIUS=0.7 and
TAU\_EM\_COLLIMATION = 0.95.
No further calorimetric isolation is required.

The tracking isolation for the tau identification requires that the number of
tracks associated to a particle with $p_T > PT\_TRACK\_TAU$ is one and only one
in a cone with $\Delta R = TAU\_CONE\_TRACKS$. This cone should be entirely
included in the tracker to be taken into account. Typical calues are
$TAU\_CONE\_TRACKS = 0.4$ and $PT\_TRACK\_TAU = 2 GeV$.

This means that the list of tau jet candidates includes the electron candidates.



\subsection{B-tagging}
The simulation of the b-tagging is based on the detector efficiencies assumed
(1) for the tagging of a b-jet and (2) for the mis-identification of other jets
as b-jets. This relies on the TAGGING\_B, MISTAGGING\_C and MISTAGGING\_L
constants, for (respectively) the efficiency of tagging of a b-jet, the
efficiency of mistagging a c-jet as a b-jet, and the efficiency of mistatting a
light jet (u,d,s,g) as a b-jet. The (mis)tagging relies on the particle ID of
the most energetic particle within a cone around the observed (eta,phi) region,
with a radius CONERADIUS.

\section{Validation}	
\section{conclusion}

\appendix
Attention : in SmearUtil::NumTracks, the function arguments 'Eta' and 'Phi' have
been switched. Previously, 'Phi' was before 'Eta', now 'Eta' comes in front.
This is for consistency with the other functions in SmearUtil. Check your
routines, when using NumTracks !

In the list of input files, all files should have the same type 

Attention : in SmearUtil::RESOLution::BJets, the maximal energy was looked in
CONERADIUS/2 instead of CONERADIUS. This bug has been removed.

Attention : for the tau-jet identification : CONERADIUS /2 was used instead of
CONERADIUS !

\end{document}
