Mior fixes
[u/mrichter/AliRoot.git] / PWG2 / FORWARD / doc / doc.tex
655b45b0 1\documentclass[11pt]{article}
ffa07380 6\usepackage{graphicx}
56bd6baf 10\usepackage[colorlinks,urlcolor=black,hyperindex,%
dc64f2ea 11 linktocpage,a4paper,bookmarks=true,%
12 bookmarksopen=true,bookmarksopenlevel=2,%
13 bookmarksnumbered=true]{hyperref}
14%% \usepackage{bookmark}
ffa07380 15\def\AlwaysText#1{\ifmmode\relax\text{#1}\else #1\fi}
16\newcommand{\AbbrName}[1]{\AlwaysText{{\scshape #1}}}
56bd6baf 17\newcommand{\CERN}{\AbbrName{cern}}
655b45b0 19\newcommand{\SPD}{\AbbrName{spd}}
8c548214 25\newcommand{\FMD}[1][]{\AbbrName{fmd\ifx|#1|\else#1\fi}}
56bd6baf 26\newcommand{\OCDB}{\AbbrName{ocdb}}
27\newcommand{\mult}[1][]{\ensuremath N_{\text{ch}#1}}
655b45b0 28\newcommand{\dndetadphi}[1][]{{\ensuremath%
29 \ifx|#1|\else\left.\fi%
56bd6baf 30 \frac{d^2\mult{}}{d\eta\,d\varphi}%
655b45b0 31 \ifx|#1|\else\right|_{#1}\fi%
34 \text{landau}\left(#1\right)}}
36 \ifx|#1|\else\left.\fi%
56bd6baf 37 \frac{1}{N}\frac{d\mult{}}{d\eta}%
655b45b0 38 \ifx|#1|\else\right|_{#1}\fi%
ffa07380 40\newcommand{\MC}{\AlwaysText{MC}}
fc6a90cc 41\newcommand{\N}[2]{{\ensuremath N_{#1#2}}}
56bd6baf 46\newcommand{\Ngood}{{\ensuremath N_{\text{good}}}}
ffa07380 47\newcommand{\GeV}[1]{\unit[#1]{\AlwaysText{GeV}}}
56bd6baf 49\newcommand{\secref}[1]{Section~\ref{#1}}
dc64f2ea 52% Azimuthal acceptance
53\newcommand{\Corners}{\ensuremath A^{\varphi}_{t}}
54% Acceptance due to dead strips
55\newcommand{\DeadCh}{\ensuremath A^{\eta}_{v,i}\etaphi}
56\newcommand{\SecMap}{\ensuremath S_v\etaphi}
655b45b0 57\setlength{\parskip}{1ex}
655b45b0 59\title{Analysing the FMD data for $\dndeta$}
60\author{Christian Holm
ffa07380 61 Christensen\thanks{\texttt{$\langle$cholm@nbi.dk$\rangle$}}\quad\&\quad
62 Hans Hjersing Dalsgaard\thanks{\texttt{$\langle$canute@nbi.dk$\rangle$}}\\
655b45b0 63 Niels Bohr Institute\\
64 University of Copenhagen}
dc64f2ea 67\pdfbookmark{Analysing the FMD data for dN/deta}{top}
655b45b0 68\maketitle
ffa07380 70\tableofcontents
655b45b0 72
73This document describes the steps performed in the analysis of the
74charged particle multiplicity in the forward pseudo--rapidity
56bd6baf 75regions. The primary detector used for this is the \FMD{}
dc64f2ea 76\cite{FWD:2004mz,cholm:2009}.
78The \FMD{} is
79organised in 3 \emph{sub--detectors} \FMD{1}, \FMD{2}, and \FMD{3}, each
80consisting of 1 (\FMD{1}) or 2 (\FMD{2} and~3) \emph{rings}.
81The rings fall into two types: \emph{Inner} or \emph{outer} rings.
82Each ring is in turn azimuthally divided into \emph{sectors}, and each
83sector is radially divided into \emph{strips}. How many sectors,
84strips, as well as the $\eta$ coverage is given in
88 \begin{center}
89 \caption{Physical dimensions of Si segments and strips.}
90 \label{tab:fmd:overview}
91 \vglue0.2cm
92 \begin{tabular}{|c|cc|cr@{\space--\space}l|r@{\space--\space}l|}
93 \hline
94 \textbf{Sub--detector/} &
95 \textbf{Azimuthal}&
96 \textbf{Radial} &
97 $z$ &
98 \multicolumn{2}{c|}{\textbf{$r$}} &
99 \multicolumn{2}{c|}{\textbf{$\eta$}} \\
100 \textbf{Ring}&
101 \textbf{sectors} &
102 \textbf{strips} &
103 \textbf{[cm]} &
104 \multicolumn{2}{c|}{\textbf{range [cm]}} &
105 \multicolumn{2}{c|}{\textbf{coverage}} \\
106 \hline
107 FMD1i & 20& 512& 320 & 4.2& 17.2& 3.68& 5.03\\
108 FMD2i & 20& 512& 83.4& 4.2& 17.2& 2.28& 3.68\\
109 FMD2o & 40& 256& 75.2& 15.4& 28.4& 1.70& 2.29\\
110 FMD3i & 20& 512& -75.2& 4.2& 17.2&-2.29& -1.70\\
111 FMD3o & 40& 256& -83.4& 15.4& 28.4&-3.40& -2.01\\
112 \hline
113 \end{tabular}
114 \end{center}
b9bd46b7 117The \FMD{} \ESD{} object contains the scaled energy deposited $\Delta
118E/\Delta E_{mip}$ for each of the 51,200 strips. This is determined
119in the reconstruction pass. The scaling to $\Delta E_{mip}$ is done
120using calibration factors extracted in designated pulser runs. In
121these runs, the front-end electronics is pulsed with an increasing
122known pulse size, and the conversion factor from ADC counts to $\Delta
123E_{mip}$ is determined \cite{cholm:2009}.
dc64f2ea 125The \SPD{} is used for determination of the position of the primary
126interaction point.
655b45b0 127
128The analysis is performed as a two--step process.
130\item The Event--Summary--Data (\ESD{}) is processed event--by--event
131 and passed through a number of algorithms, and
132 $\dndetadphi$ for each event is output to an Analysis--Object--Data
dc64f2ea 133 (\AOD{}) tree (see \secref{sec:gen_aod}).
655b45b0 134\item The \AOD{} data is read in and the sub--sample of the data under
135 investigation is selected (e.g., \INEL{}, \INELONE{}, \NSD{}, or
136 some centrality class) and the $\dndetadphi$ histogram read in for
dc64f2ea 137 those events to build up $\dndeta$ (see \secref{sec:ana_aod}).
655b45b0 138\end{enumerate}
139The details of each step above will be expanded upon in the
dc64f2ea 142In Appendix~\ref{app:nomen} is an overview of the nomenclature used in
143this document.
ffa07380 147\section{Generating $\dndetadphi[i]$ event--by--event}
dc64f2ea 148\label{sec:gen_aod}
655b45b0 149
150When reading in the \ESD{}s and generating the $\dndetadphi$
151event--by--event the following steps are taken (in order) for each
152event $i$
154\item[Event inspection] The global properties of the event is
56bd6baf 155 determined, including the trigger type and primary interaction
156 point\footnote{`Vertex' and `primary interaction point' will be used
157 interchangeably in the text, since there is no ambiguity with
158 particle production vertex in this analysis.} $z$ coordinate (see
159 \secref{sec:sub:event_inspection}).
655b45b0 160\item[Sharing filter] The \ESD{} object is read in and corrected for
56bd6baf 161 sharing. The result is a new \ESD{} object (see
162 \secref{sec:sub:sharing_filter}).
655b45b0 163\item[Density calculator] The (possibly un--corrected) \ESD{} object
56bd6baf 164 is then inspected and an inclusive (primary \emph{and} secondary
165 particles), per--ring charged particle density
166 $\dndetadphi[incl,r,v,i]$ is made. This calculation depends in
167 general upon the interaction vertex position along the $z$ axis
168 $v_z$ (see \secref{sec:sub:density_calculator}).
169\item[Corrections] The 5 $\dndetadphi[incl,r,v,i]$ are corrected for
170 secondary production and acceptance. The correction for the
171 secondary particle production is highly dependent on the vertex $z$
172 coordinate. The result is a per--ring, charged primary particle
173 density $\dndetadphi[r,v,i]$ (see \secref{sec:sub:corrector}).
655b45b0 174\item[Histogram collector] Finally, the 5 $\dndetadphi[r,v,i]$ are
175 summed into a single $\dndetadphi[v,i]$ histogram, taking care of
176 the overlaps between the detector rings. In principle, this
177 histogram is independent of the vertex, except that the
178 pseudo--rapidity range, and possible holes in that range, depends on
56bd6baf 179 $v_z$ --- or rather the bin in which the $v_z$ falls (see
180 \secref{sec:sub:hist_collector}).
655b45b0 181\end{description}
183Each of these steps will be detailed in the following.
ffa07380 185\subsection{Event inspection}
56bd6baf 186\label{sec:sub:event_inspection}
655b45b0 187
188The first thing to do, is to inspect the event for triggers. A number
189of trigger bits, like \INEL{}, \INELONE{}, \NSD{}, and so on is then
190propagated to the \AOD{} output.
b9bd46b7 192Just after the sharing filter (described below) but before any further
655b45b0 193processing, the vertex information is queried. If there is no vertex
194information, or if the vertex $z$ coordinate is outside the
56bd6baf 195pre--defined range, then no further processing of that event takes place.
655b45b0 196
ffa07380 197\subsection{Sharing filter}
56bd6baf 198\label{sec:sub:sharing_filter}
655b45b0 199
dc64f2ea 200A particle originating from the vertex can, because of its incident
56bd6baf 201angle on the \FMD{} sensors traverse more than one strip (see
202\figref{fig:share_fraction}). This means that the energy loss of the
203particle is distributed over 1 or more strips. The signal in each
b9bd46b7 204strip should therefore possibly be merged with its neighboring strip
56bd6baf 205signals to properly reconstruct the energy loss of a single particle.
655b45b0 206
56bd6baf 207\begin{figure}[htbp]
208 \centering
209 \includegraphics[keepaspectratio,height=3cm]{share_fraction}
210 \caption{A particle traversing 2 strips and depositing energy in
211 each strip. }
212 \label{fig:share_fraction}
215The effect is most pronounced in low--flux\footnote{Events with a low
216 hit density.} events, like proton--proton collisions or peripheral
217Pb--Pb collisions, while in high--flux events the hit density is so
218high that most likely each and every strip will be hit and the effect
219cancel out on average.
655b45b0 220
221Since the particles travel more or less in straight lines toward the
dc64f2ea 222\FMD{} sensors, the sharing effect is predominantly in the $r$ or
223\emph{strip} direction. Only neighbouring strips in a given sector is
655b45b0 224therefor investigated for this effect.
226Algorithm~\ref{algo:sharing} is applied to the signals in a given
dc64f2ea 230 \belowpdfbookmark{Algorithm 1}{algo:sharing}
655b45b0 231 \SetKwData{usedThis}{current strip used}
232 \SetKwData{usedPrev}{previous strip used}
233 \SetKwData{Output}{output}
234 \SetKwData{Input}{input}
235 \SetKwData{Nstr}{\# strips}
236 \SetKwData{Signal}{current}
237 \SetKwData{Eta}{$\eta$}
238 \SetKwData{prevE}{previous strip signal}
239 \SetKwData{nextE}{next strip signal}
240 \SetKwData{lowFlux}{low flux flag}
241 \SetKwFunction{SignalInStrip}{SignalInStrip}
242 \SetKwFunction{MultiplicityOfStrip}{MultiplicityOfStrip}
243 \usedThis $\leftarrow$ false\;
244 \usedPrev $\leftarrow$ false\;
245 \For{$t\leftarrow1$ \KwTo \Nstr}{
246 \Output${}_t\leftarrow 0$\;
247 \Signal $\leftarrow$ \SignalInStrip($t$)\;
249 \uIf{\Signal is not valid}{
250 \Output${}_t \leftarrow$ invalid\;
251 }
252 \uElseIf{\Signal is 0}{
253 \Output${}_t \leftarrow$ 0\;
254 }
255 \Else{
256 \Eta$\leftarrow$ $\eta$ of \Input${}_t$\;
257 \prevE$\leftarrow$ 0\;
258 \nextE$\leftarrow$ 0\;
259 \lIf{$t \ne 1$}{
260 \prevE$\leftarrow$ \SignalInStrip($t-1$)\;
261 }
262 \lIf{$t \ne $\Nstr}{
263 \nextE$\leftarrow$ \SignalInStrip($t+1$)\;
264 }
265 \Output${}_t\leftarrow$
266 \MultiplicityOfStrip(\Signal,\Eta,\prevE,\nextE,\\
267 \hfill\lowFlux,$t$,\usedPrev,\usedThis)\;
268 }
269 }
270 \caption{Sharing correction}
271 \label{algo:sharing}
274Here the function \FuncSty{SignalInStrip}($t$) returns the properly
275path--length corrected signal in strip $t$. The function
56bd6baf 276\FuncSty{MultiplicityOfStrip} is where the real processing takes
277place (see page \pageref{func:MultiplicityOfStrip}).
655b45b0 278
dc64f2ea 280 \belowpdfbookmark{MultiplicityOfStrip}{func:MultiplicityOfStrip}
56bd6baf 281 \caption{MultiplicityOfStrip(\DataSty{current},$\eta$,\DataSty{previous},\DataSty{next},\DataSty{low
655b45b0 282 flux flag},\DataSty{previous signal used},\DataSty{this signal
283 used})}
56bd6baf 284 \label{func:MultiplicityOfStrip}
655b45b0 285 \SetKwData{Current}{current}
286 \SetKwData{Next}{next}
287 \SetKwData{Previous}{previous}
288 \SetKwData{lowFlux}{low flux flag}
289 \SetKwData{usedPrev}{previous signal used}
290 \SetKwData{usedThis}{this signal used}
291 \SetKwData{lowCut}{low cut}
292 \SetKwData{total}{Total}
293 \SetKwData{highCut}{high cut}
294 \SetKwData{Eta}{$\eta$}
295 \SetKwFunction{GetHighCut}{GetHighCut}
296 \If{\Current is very large or \Current $<$ \lowCut} {
297 \usedThis $\leftarrow$ false\;
298 \usedPrev $\leftarrow$ false\;
299 \Return{0}
300 }
301 \If{\usedThis}{
302 \usedThis $\leftarrow$ false\;
303 \usedPrev $\leftarrow$ true\;
304 \Return{0}
305 }
306 \highCut $\leftarrow$ \GetHighCut($t$,\Eta)\;
dc64f2ea 307 %\If{\Current $<$ \Next and \Next $>$ \highCut and \lowFlux set}{
308 % \usedThis $\leftarrow$ false\;
309 % \usedPrev $\leftarrow$ false\;
310 % \Return{0}
311 %}
655b45b0 312 \total $\leftarrow$ \Current\;
313 \lIf{\lowCut $<$ \Previous $<$ \highCut and not \usedPrev}{
314 \total $\leftarrow$ \total + \Previous\;
315 }
316 \If{\lowCut $<$ \Next $<$ \highCut}{
317 \total $\leftarrow$ \total + \Next\;
318 \usedThis $\leftarrow$ true\;
319 }
320 \eIf{\total $>$ 0}{
321 \usedPrev $\leftarrow$ true\;
322 \Return{\total}
323 }{
324 \usedPrev $\leftarrow$ false\;
325 \usedThis $\leftarrow$ false\;
326 \Return{0}
327 }
56bd6baf 329Here, the function \FuncSty{GetHighCut} evaluates a fit to the energy
330distribution in the specified $\eta$ bin (see also
331\secref{sec:sub:density_calculator}). It returns
655b45b0 332$$
333\Delta_{mp} - 2 w
335where $\Delta_{mp}$ is the most probable energy loss, and $w$ is the
336width of the Landau distribution.
338The \KwSty{if} in line 5, says that if the previous strip was merged
339with current one, and the signal of the current strip was added to
56bd6baf 340that, then the current signal is set to 0, and we mark it as used for
341the next iteration (\DataSty{previous signal used}$\leftarrow$true).
655b45b0 342
dc64f2ea 343% The \KwSty{if} in line 10 checks if the current signal is smaller than
344% the next signal, if the next signal is larger than the upper cut
345% defined above, and if we have a low--flux event\footnote{Note, that in
346% the current implementation there are never any low--flux events.}.
347% If that condition is met, then the current signal is the smaller of
348% two possible candidates for merging, and it should be merged into the
349% next signal. Note, that this \emph{only} applies in low--flux events.
56bd6baf 350
dc64f2ea 351In line 11, % 15,
352we test if the previous signal lies between our low and
655b45b0 353high cuts, and if it has not been marked as being used. If so, we add
354it to our current signal.
dc64f2ea 356The next \KwSty{if} on line 12 % 16
357checks if the next signal is within our
655b45b0 358cut bounds. If so, we add that signal to the current signal and mark
359it as used for the next iteration (\DataSty{this signal
360 used}$\leftarrow$true). It will then be zero'ed on the next
361iteration by the condition on line 6.
363Finally, if our signal is still larger than 0, we return the signal
364and mark this signal as used (\DataSty{previous signal
365 used}$\leftarrow$true) so that it will not be used in the next
366iteration. Otherwise, we mark the current signal and the next signal
367as unused and return a 0.
ffa07380 370\subsection{Density calculator}
56bd6baf 371\label{sec:sub:density_calculator}
655b45b0 372
dc64f2ea 373The density calculator loops over all the strip signals in the sharing
374corrected\footnote{The sharing correction can be disabled, in which
375 case the density calculator used the input \ESD{} signals.} \ESD{}
56bd6baf 376and calculates the inclusive (primary + secondary) charged particle
377density in pre--defined $\etaphi$ bins.
655b45b0 378
ffa07380 379\subsubsection{Inclusive number of charged particles}
dc64f2ea 380\label{sec:sub:sub:eloss_fits}
ffa07380 381
b9bd46b7 382The number charged particles in a strip $\mult[,t]$ is calculated
383using multiple Landau-like distributions fitted to the energy loss
384spectrum of all strips in a given at a given $\eta$ bin.
655b45b0 385\begin{align}
0a89eed1 386 \Delta_{i,mp} &= i (\Delta_{1,mp}+ \xi_1 \log(i))\nonumber\\
387 \xi_i &= i\xi_1\nonumber\\
388 \sigma_i &= \sqrt{i}\sigma_1\nonumber\\
56bd6baf 389 \mult[,t] &= \frac{\sum_i^{N_{max}}
0a89eed1 390 i\,a_i\,F(\Delta_t;\Delta_{i,mp},\xi_i,\sigma_i)}{
391 \sum_i^{N_{max}}\,a_i\,F(\Delta_t;\Delta_{i,mp},\xi_i,\sigma_i)}\quad,
655b45b0 392\end{align}
0a89eed1 393where $F(x;\Delta_{mp},\xi,\sigma)$ is the evaluation of the Landau
394distribution $f_L$ with most probable value $\Delta_{mp}$ and width
56bd6baf 395$\xi$, folded with a Gaussian distribution with spread $\sigma$ at the
396energy loss $x$ \cite{nim:b1:16,phyrev:a28:615}.
398 \label{eq:energy_response}
399 F(x;\Delta_{mp},\xi,\sigma) = \frac{1}{\sigma \sqrt{2 \pi}}
400 \int_{-\infty}^{+\infty} d\Delta' f_{L}(x;\Delta',\xi)
401 \exp{-\frac{(\Delta_{mp}-\Delta')^2}{2\sigma^2}}\quad,
403where $\Delta_{1,mp}$, $\xi_1$, and $\sigma_1$ are the parameters for
404the first MIP peak, $a_1=1$, and $a_i$ is the relative weight of the
dc64f2ea 405$i$-fold MIP peak. The parameters $\Delta_{1,mp}, \xi_1,
406\sigma_1, \mathbf{a} = \left(a_2, \ldots a_{N_{max}}\right)$ are
407obtained by fitting
0a89eed1 408$$
dc64f2ea 409F_j(x;C,\Delta_{mp},\xi,\sigma,\mathbf{a}) = C
410\sum_{i=1}^{j} a_i F(x;\Delta_{i,mp},\xi_{i},\sigma_i)
0a89eed1 411$$
56bd6baf 412for increasing $j$ to the energy loss spectra in separate $\eta$ bins.
b9bd46b7 413The fit procedure is stopped when one for $j+1$
415\item the reduced $\chi^2$ exceeds a certain threshold, or
416\item the relative error $\delta p/p$ of any parameter of the fit
417 exceeds a certain threshold, or
418\item when the weight $a_j+1$ is smaller than some number (typically
419 $10^{-5}$).
421$N_{max}$ is then set to $j$. Examples of the result of these fits
422are given in \figref{fig:eloss_fits} in Appendix~\ref{app:eloss_fits}.
655b45b0 423
56bd6baf 424\subsubsection{Azimuthal Acceptance}
ffa07380 425
56bd6baf 426Before the signal $\mult[,t]$ can be added to the $\etaphi$
655b45b0 427bin in one of the 5 per--ring histograms, it needs to be corrected for
56bd6baf 428the $\varphi$ acceptance of the strip.
655b45b0 429
b9bd46b7 430The sensors of the \FMD{} are not perfect arc--segments --- the two
431top corners are cut off to allow the largest possible sensor on a 6''
432Si-wafer. This means, however, that the strips in these outer
433regions do not fully cover $2\pi$ in azimuth, and there is therefore a
434need to correct for this limited acceptance.
655b45b0 436The acceptance correction is only applicable where the strip length
437does not cover the full sector. This is the case for the outer strips
438in both the inner and outer type rings. The acceptance correction is
439then simply
441 \label{eq:acc_corr}
dc64f2ea 442 \Corners{} &= \frac{l_t}{\Delta\varphi}\quad
655b45b0 443\end{align}
444where $l_t$ is the strip length in radians at constant $r$, and
445$\Delta\varphi$ is $2\pi$ divided by the number of sectors in the
446ring (20 for inner type rings, and 40 for outer type rings).
b9bd46b7 448Note, that this correction is a hardware--related correction, and does
449not depend on the properties of the collision (e.g., primary vertex
56bd6baf 452The final $\etaphi$ content of the 5 output vertex dependent,
655b45b0 453per--ring histograms of the inclusive charged particle density is then
454given by
8c548214 456 \label{eq:density}
56bd6baf 457 \dndetadphi[incl,r,v,i\etaphi] &= \sum_t^{t\in\etaphi}
dc64f2ea 458 \mult[,t]\,\Corners{}
655b45b0 459\end{align}
56bd6baf 460where $t$ runs over the strips in the $\etaphi$ bin.
655b45b0 461
ffa07380 462\subsection{Corrections}
56bd6baf 463\label{sec:sub:corrector}
655b45b0 464
465The corrections code receives the five vertex dependent,
466per--ring histograms of the inclusive charged particle density
467$\dndetadphi[incl,r,v,i]$ from the density calculator and applies
56bd6baf 468two corrections
ffa07380 469
470\subsubsection{Secondary correction}
472%% hHits_FMD<d><r>_vtx<v>
473%% hCorrection = -----------------------
474%% hPrimary_FMD_<r>_vtx<v>
476%% where
477%% - hPrimary_FMD_<r>_vtx<vtx> is 2D of eta,phi for all primary ch
478%% particles
479%% - hHits_FMD<d><r>_vtx<v> is 2D of eta,phi for all track-refs that
480%% hit the FMD - The 2D version of hMCHits_nocuts_FMD<d><r>_vtx<v>
481%% used below.
56bd6baf 482This is a 2 dimensional histogram generated from simulations, as the
483ratio of primary particles to the total number of particles that fall
484within an $\etaphi$ bin for a given vertex bin
487 \label{eq:secondary}
dc64f2ea 488 \SecMap{} &=
fc6a90cc 489 \frac{\sum_i^{\NV[,v]}\mult[,\text{primary},i]\etaphi}{
490 \sum_i^{\NV[,v]}\mult[,\text{\FMD{}},i]\etaphi}\quad,
56bd6baf 491\end{align}
fc6a90cc 492where $\NV[,v]$ is the number of events with a valid trigger and a
56bd6baf 493vertex in bin $v$, and $\mult[,\FMD{},i]$ is the total number of
494charged particles that hit the \FMD{} in event $i$ in the specified
495$\etaphi$ bin and $\mult[,\text{primary},i]$ is number of
496primary charged particles in event $i$ within the specified
497$\etaphi$ bin.
499$\mult[,\text{primary}]\etaphi$ is given by summing over the
500charged particles labelled as primaries \emph{at the time of the
501 collision} as defined in the simulation code. That is, it is the
502number of primaries within the $\etaphi$ bin at the collision
503point --- not at the \FMD{}.
b9bd46b7 505$\SecMap$ is varies from $\approx 1.5$ for the most forward bins to
506$\approx 3$ for the more central bins. For pp, different event
507generators were used and found to give compatible results within
5083--5\%. For pp, at least some millions of events must be
509accumulated to reach satisfactory statistics. For Pb--Pb where the
510general hit density is larger, reasonable statistics can be achieved
511with less data.
56bd6baf 513\subsubsection{Acceptance due to dead channels}
515Some of the strips in the \FMD{} have been marked up as \emph{dead},
516meaning that they are not used in the reconstruction or analysis.
517This leaves holes in the acceptance of each defined $\etaphi$
518which need to be corrected for.
520Dead channels are marked specially in the \ESD{}s with the flag
521\textit{Invalid Multiplicity}. This is used in the analysis to build
522up and event--by--event acceptance correction in each $\etaphi$
523bin by calculating the ratio
ffa07380 524\begin{align}
56bd6baf 525 \label{eq:dead_channels}
dc64f2ea 526 \DeadCh{} &=
56bd6baf 527 \frac{\sum_t^{t\in\etaphi}\left\{\begin{array}{cl}
528 1 & \text{if not dead}\\
529 0 & \text{otherwise}
530 \end{array}\right.}{\sum_t^{t\in\etaphi} 1}\quad,
ffa07380 531\end{align}
dc64f2ea 532where $t$ runs over the strips in the $\etaphi$ bin. This correction
533is obviously $v_z$ dependent since which $\etaphi$ bin a strip $t$
534corresponds to depends on its relative position to the primary vertex.
56bd6baf 535
536Alternatively, pre--made acceptance factors can be used. These are
537made from the off-line conditions database (\OCDB{}).
655b45b0 538
539The 5 output vertex dependent, per--ring histograms of the primary
540charged particle density is then given by
56bd6baf 542 \dndetadphi[r,v,i\etaphi] &=
dc64f2ea 543 \SecMap{} \frac{1}{\DeadCh{}}\dndetadphi[incl,r,v,i\etaphi]
655b45b0 544\end{align}
ffa07380 546\subsection{Histogram collector}
56bd6baf 547\label{sec:sub:hist_collector}
655b45b0 548
549The histogram collector collects the information from the 5 vertex
550dependent, per--ring histograms of the primary charged particle
551density $\dndetadphi[r,v,i]$ into a single vertex dependent histogram
552of the charged particle density $\dndetadphi[v,i]$.
554To do this, it first calculates, for each vertex bin, the $\eta$ bin
555range to use for each ring. It investigates the secondary correction
dc64f2ea 556maps $\SecMap{}$ to find the edges of each map. The edges are given
557by the $\eta$ range where $\SecMap{}$ is larger than some
558threshold\footnote{Typically $t_s\approx 0.1$.} $t_s$. The code
8c548214 559applies safety margin of a $N_{cut}$ bins\footnote{Typically
560 $N_{cut}=1$.}, to ensure that the data selected does not have too
561large corrections associated with it.
655b45b0 562
563It then loops over the bins in the defined $\eta$ range and sums the
8c548214 564contributions from each of the 5 histograms. In the $\eta$ ranges
565where two rings overlap, the collector calculates the average and adds
b9bd46b7 566the errors in quadrature\footnote{While not explicitly checked, it was
567 found that the histograms agrees within error bars in the
568 overlapping region}.
655b45b0 569
570The output vertex dependent histogram of the primary
571charged particle density is then given by
ffa07380 573 \label{eq:superhist}
56bd6baf 574 \dndetadphi[v,i\etaphi] &=
575 \frac{1}{N_{r\in\etaphi}}\sum_{r}^{r\in\etaphi}
576 \dndetadphi[r,v,i\etaphi]\\
577 \delta\left[\dndetadphi[v,i\etaphi]\right] &=
578 \frac{1}{N_{r\in\etaphi}}\sqrt{\sum_{r}^{r\in\etaphi}
579 \delta\left[\dndetadphi[r,v,i\etaphi]\right]^2}
655b45b0 580 \quad,
56bd6baf 582where $N_{r\in\etaphi}$ is the number of overlapping histograms
583in the given $\etaphi$ bin.
655b45b0 584
ffa07380 585The histogram collector stores the found $\eta$ ranges in the
586underflow bin of the histogram produced. The content of the overflow
587bins are
589 \label{eq:overflow}
590 I_{v,i}(\eta) &=
591 \frac{1}{N_{r\in(\eta)}}
592 \sum_{r}^{r\in(\eta)} \left\{\begin{array}{cl}
593 0 & \eta \text{\ bin not selected}\\
594 1 & \eta \text{\ bin selected}
595 \end{array}\right.\quad,
597where $N_{r\in(\eta)}$ is the number of overlapping histograms in the
598given $\eta$ bin. The subscript $v$ indicates that the content
599depends on the current vertex bin of event $i$.
601\section{Building the final $\dndeta$}
dc64f2ea 602\label{sec:ana_aod}
ffa07380 603
604To build the final $\dndeta$ distribution it is enough to sum
605\eqref{eq:superhist} and \eqref{eq:overflow} over all interesting
fc6a90cc 606events and correct for the acceptance $I(\eta)$
56bd6baf 607\begin{align}
fc6a90cc 608 \dndetadphi[\etaphi] &= \sum_i^{\NA}\dndetadphi[i,v\etaphi]\\
609 I(\eta) &= \sum_i^{\NA}I_{i,v}(\eta)\quad.
56bd6baf 610\end{align}
fc6a90cc 611Note, that $I(\eta)\le\NA$.
56bd6baf 612
fc6a90cc 613We then need to normalise to the total number of events $N_X$, given
ffa07380 615\begin{align}
fc6a90cc 616 \N{X}{} &= \frac{1}{\epsilon_X}\left[\NA + \alpha(\NnotV -
617 \beta)\right] \label{eq:fulleventnorm}\\
618 & = \frac{1}{\epsilon_X}\left[\NA + \frac{\NA}{\NV}(\NT-\NV{} -
619 \beta)\right]\nonumber \\
620 & =\frac{1}{\epsilon_X}\NA\left[1+\frac{1}{\epsilon_V}-1-
621 \frac{\beta}{\NV}\right]\nonumber\\
622 & = \frac{1}{\epsilon_X}\frac{1}{\epsilon_V}\NA
623 \left(1-\frac{\beta}{\NT{}}\right)\nonumber
627\item[$\epsilon_X$] is the trigger efficiency for type
628 $X\in[\text{\INEL},\text{\INELONE},\text{\NSD},...]$
629\item[$\epsilon_V=\frac{\NV{}}{\NT{}}$] is the vertex efficiency
630 evaluated over the data.
631\item[$\NA$] is the number of events with a trigger \emph{and} a valid
632 vertex in the selected range
633\item[$\NV{}$] is the number of events with a trigger \emph{and} a valid
634 vertex.
635\item[$\NT$] is the number of events with a trigger.
636\item[$\NnotV{}=\NT-\NV{}$] is the number of events with a trigger
637 \emph{but no} valid vertex
638\item[$\alpha=\frac{\NA}{\NV}$] is the fraction of accepted events of
639 the total number of events with a trigger and valid vertex.
640\item[$\beta=\N{a}{}+\N{b}{}-\N{e}{}$] is the number of background
641 events \emph{with} a valid off-line trigger.
643The two terms under the parenthesis in \eqref{eq:fulleventnorm} refers
644to the observed number of event $\NA$, and the events missed because
645of no vertex reconstruction. Note, for $\beta\ll\NT{}$
646\eqref{eq:fulleventnorm} reduces to the simpler expression
648\N{X}{} = \frac1{\epsilon_X}\frac1{\epsilon_V}\NA{}
650The trigger efficiency $\epsilon_X$ for a given trigger type $X$ is
651evaluated from simulations as
653 \epsilon_X = \frac{\N{X\wedge \text{T}}{}}{\N{X}{}}\quad,
655that is, the ratio of number of events of type $X$ with a
656corresponding trigger to the number of events of type $X$.
658The final event--normalised charged particle density then becomes
660 \frac{1}{N}\frac{dN_{\text{ch}}}{d\eta} &=
661 \frac{1}{\N{X}{}} \int_0^{2\pi} d\varphi
662 \frac{\dndetadphi[\etaphi]}{I(\eta)}
663 \label{eq:eventnormdndeta}
666If the trigger $X$ introduces a bias on the measured number of events,
667then \eqref{eq:eventnormdndeta} need to be modified to
669 \frac{1}{N}\frac{dN_{\text{ch}}}{d\eta} &=
670 \frac{1}{\N{X}{}} \int_0^{2\pi} d\varphi
671 \frac{\frac{1}{B\etaphi}\dndetadphi[\etaphi]}{I(\eta)}
672 \label{eq:eventnormdndeta2}\quad,
674where $B\etaphi$ is the bias correction. This is typically
675calculated from simulations using the expression
677 B\etaphi = \frac{\frac{1}{\N{X\wedge
678 \text{T}}{}}\sum_i^{\N{X\wedge \text{T}}{}}
679 \mult[,\text{primary}]\etaphi}{\frac{1}{\N{X}{}}\sum_i^{\N{X}{}}
680 \mult[,\text{primary}]\etaphi}
ffa07380 681\end{align}
fc6a90cc 682
655b45b0 683
ffa07380 684\section{Using the per--event $\dndetadphi[i,v]$ histogram for other
685 analysis}
655b45b0 686
ffa07380 687\subsection{Multiplicity distribution}
655b45b0 688
ffa07380 689To build the multiplicity distribution for a given $\eta$ range
690$[\eta_1,\eta_2]$, one needs to find the total multiplicity in that
691$\eta$ range for each event. To do so, one should sum the
692$\dndetadphi[i,v]$ histogram over all $\varphi$ and in the selected
693$\eta$ range.
695 n'_{i[\eta_1,\eta_2]}, &= \int_{\eta_1}^{\eta_2}d\eta\int_0^{2\pi}d\varphi
696 \dndetadphi[i,v]\quad.\nonumber
698However, $n'_i$ is not corrected for the coverage in $\eta$ for the
699particular vertex range $v$. One therefor needs to correct for the
700number of missing bins in the range $[\eta_1,\eta_2]$. Suppose
701$[\eta_1,\eta_2]$ covers $N_{[\eta_1,\eta_2]}$ $\eta$ bins, then the acceptance
702correction is given by
704 A_{i,[\eta_1,\eta_2]} = \frac{N_{[\eta_1,\eta_2]}}{\int_{\eta_1}^{\eta_2}d\eta\,
705 I_{i,v}(\eta)}\quad.\nonumber
707The per--event multiplicity is then given by
709 n_{i,[\eta_1,\eta_2]} &= A_{i,[\eta_1,\eta_2]}\,n'_{i,[\eta_1,\eta_2]}\nonumber\\
710 &= \frac{N_{[\eta_1,\eta_2]}}{\int_{\eta_1}^{\eta_2}\eta
711 I_{i,v}(\eta)} \int_{\eta_1}^{\eta_2}d\eta\int_0^{2\pi}d\varphi
712 \dndetadphi[i,v]
713 \label{eq:event_n}
716\subsection{Forward--Backward correlations}
718To do forward--backward correlations, one need to calculate
719$n_{i,[\eta_1,\eta_2]}$ as shown in \eqref{eq:event_n} in two bins
720$n_{i,[\eta_1,\eta_2]}$ and $n_{i,[-\eta_2,-\eta_1]}$ \textit{e.g.},
721$n_{i,f}=n_{i,[-3,-1]}$ and $n_{i,b}=n_{i,[1,3]}$.
dc64f2ea 723\clearpage
ffa07380 724\section{Some results}
dc64f2ea 726%% \figurename{}s \ref{fig:1} to \ref{fig:3} shows some results.
727Figures below show some examples. Note these are not finalised
ffa07380 729
dc64f2ea 730\begin{figure}[hbp]
ffa07380 731 \centering
732 \includegraphics[keepaspectratio,width=\textwidth]{%
b9bd46b7 733 dndeta_pp_0900GeV_INEL_m10p10cm}
ffa07380 734 \caption{$\dndeta$ for pp for \INEL{} events at $\sqrt{s}=\GeV{900}$,
b9bd46b7 735 $\cm{-10}\le v_z\le\cm{10}$, rebinned by a factor 5. Middle panel
736 shows the ratio of ALICE data to UA5, and the bottom panel shows
737 the ratio of the right (positive) side to the left (negative) side
738 of the forward $\dndeta$.}
ffa07380 739 \label{fig:1}
dc64f2ea 741
ffa07380 743\begin{figure}[tbp]
744 \centering
745 \includegraphics[keepaspectratio,width=\textwidth]{%
746 dndeta_0900GeV_m10-p10cm_rb05_inelgt0}
747 \caption{$\dndeta$ for pp for \INELONE{} events at
748 $\sqrt{s}=\GeV{900}$, $\cm{-10}\le v_z\le\cm{10}$, rebinned by a
749 factor 5. Comparisons to other measurements shown where
750 applicable}
751 \label{fig:2}
754 \centering
755 \includegraphics[keepaspectratio,width=\textwidth]{%
756 dndeta_0900GeV_m10-p10cm_rb05_nsd}
757 \caption{$\dndeta$ for pp for \NSD{} events at $\sqrt{s}=\GeV{900}$,
758 $\cm{-10}\le v_z\le\cm{10}$, rebinned by a factor 5. Comparisons
759 to other measurements shown where applicable}
760 \label{fig:3}
dc64f2ea 762\fi
ffa07380 763
dc64f2ea 765%% \currentpdfbookmark{Appendices}{Appendices}
ffa07380 766\appendix
56bd6baf 767\section{Nomenclature}
dc64f2ea 768\label{app:nomen}
56bd6baf 769
771 \centering
772 \begin{tabular}[t]{|lp{.8\textwidth}|}
773 \hline
774 \textbf{Symbol}&\textbf{Description}\\
775 \hline
776 \INEL & In--elastic event\\
777 \INELONE & In--elastic event with at least one tracklet in the
778 \SPD{} in the region $-1\le\eta\le1$\\
779 \NSD{} & Non--single--diffractive event. Single diffractive
780 events are events where one of the incident collision systems
781 (proton or nucleus) is excited and radiates particles, but there
782 is no other processes taking place\\
783 \hline
fc6a90cc 784 $\NT{}$ & Number of events with a valid trigger\\
785 $\NV{}$ & Number of events with a valid trigger \emph{and} a valid
786 vertex.\\
787 $\NA{}$ & Number of events with a valid trigger
788 \emph{and} a valid vertex \emph{within} the selected vertex range.\\
789 $\N{a,c,ac,e}{}$ & Number of events with background triggers $A$,
790 $B$, $AC$, or $E$, \emph{and} a valid off-line trigger of the
791 considered type. Background triggers are typically flagged with
792 the trigger words \texttt{CINT1-A}, \texttt{CINT1-C},
793 \texttt{CINT1-AC}, \texttt{CINT1-E}, or similar.\\
56bd6baf 794 \hline
795 $\mult{}$ & Charged particle multiplicity\\
796 $\mult[,\text{primary}]$ & Primary charged particle multiplicity
797 as given by simulations\\
798 $\mult[,\text{\FMD{}}]$ & Number of charged particles that hit the
799 \FMD{} as given by simulations\\
800 $\mult[,t]$ & Number of charged particles in an \FMD{} strip as
801 given by evaluating the energy response functions $F$\\
802 \hline
803 $F$ & Energy response function (see \eqref{eq:energy_response})\\
804 $\Delta_{mp}$ & Most probably energy loss\\
805 $\xi$ & `Width' parameter of a Landau distribution\\
806 $\sigma$ & Variance of a Gaussian distribution\\
dc64f2ea 807 $a_i$ & Relative weight of the $i$--fold MIP peak in the energy
56bd6baf 808 loss spectra.\\
809 \hline
dc64f2ea 810 $\Corners{}$ & Azimuthal acceptance of strip $t$\\
811 $\SecMap{}$ & Secondary particle correction factor in $\etaphi$
812 for a given vertex bin $v$\\
813 $\DeadCh{}$ & Acceptance in $\etaphi$ for a given vertex bin $v$\\
56bd6baf 814 \hline
815 $\dndetadphi[incl,r,v,i]$ & Inclusive (primary \emph{and}
816 secondary) charge particle density in event $i$ with vertex $v$,
817 for \FMD{} ring $r$.\\
818 $\dndetadphi[r,v,i]$ & Primary charged particle
819 density in event $i$ with vertex $v$ for \FMD{} ring $r$. \\
820 $\dndetadphi[v,i]$ & Primary charged particle density in event $i$
821 with vertex $v$\\
822 $I_{v,i}(\eta)$ & $\eta$ acceptance of event $i$ with vertex $v$\\
fc6a90cc 823 $I(\eta)$ & Integrated $\eta$ acceptance over $\NA$ events.
824 Note, that this has a value of $\NA$ for $(\eta)$ bins where we
56bd6baf 825 have full coverage\\
826 \hline
b9bd46b7 827 $X_t$ & Value $X$ for strip number $t$ (0-511 for inner rings,
828 0-255 for outer rings)\\
829 $X_r$ & Value $X$ for ring $r$ (where rings are \FMD{1i},
830 \FMD{2i}, \FMD{2o}, \FMD{3o}, and \FMD{3i} in decreasing $\eta$
831 coverage).\\
832 $X_v$ & Value $X$ for vertex bin $v$ (typically 10 bins from -10cm
833 to +10cm)\\
834 $X_i$ & Value $X$ for event $i$\\
835 \hline
56bd6baf 836 \end{tabular}
837 \caption{Nomenclature used in this document}
838 \label{tab:nomenclature}
ffa07380 843\section{Second pass example code}
56bd6baf 844\label{app:exa_pass2}
ffa07380 845\lstset{basicstyle=\small\ttfamily,%
846 keywordstyle=\color[rgb]{0.627,0.125,0.941}\bfseries,%
847 identifierstyle=\color[rgb]{0.133,0.545,0.133}\itshape,%
848 commentstyle=\color[rgb]{0.698,0.133,0.133},%
849 stringstyle=\color[rgb]{0.737,0.561,0.561},
fc6a90cc 850 emph={TH2D,TH1D,TFile,TTree,AliAODForwardMult},emphstyle=\color{blue},%
ffa07380 851 emph={[2]dndeta,sum,norm},emphstyle={[2]\bfseries\underbar},%
fc6a90cc 852 emph={[3]file,tree,mult,nV,nBg,nA,nT,i,gSystem},emphstyle={[3]},%
ffa07380 853 language=c++,%
855\begin{lstlisting}[caption={Example 2\textsuperscript{nd} pass code to
856 do $\dndeta$},label={lst:example},frame=single,captionpos=b]
fc6a90cc 857void Analyse(int mask=AliAODForwardMult::kInel,
858 float vzLow=-10, float vzHigh=10, float trigEff=1)
ffa07380 859{
860 gSystem->Load("libANALYSIS.so"); // Load analysis libraries
861 gSystem->Load("libANALYSISalice.so"); // General ALICE stuff
56bd6baf 862 gSystem->Load("libPWG2forward2.so"); // Forward analysis code
ffa07380 863
fc6a90cc 864 int nT = 0; // # of ev. w/trigger
865 int nV = 0; // # of ev. w/trigger&vertex
866 int nA = 0; // # of accepted ev.
867 int nBg = 0; // # of background ev
868 TH2D* sum = 0; // Summed hist
869 AliAODForwardMult* mult = 0; // AOD object
870 TFile* file = TFile::Open("AliAODs.root","READ");
871 TTree* tree = static_cast<TTree*>(file->Get("aodTree"));
872 tree->SetBranchAddress("Forward", &forward); // Set the address
ffa07380 873
56bd6baf 874 for (int i = 0; i < tree->GetEntries(); i++) {
ffa07380 875 // Read the i'th event
876 tree->GetEntry(i);
878 // Create sum histogram on first event - to match binning to input
0a89eed1 879 if (!sum)
880 sum = static_cast<TH2D*>(mult->GetHistogram()->Clone("d2ndetadphi"));
ffa07380 881
fc6a90cc 882 // Calculate beta=A+C-E
883 if (mult->IsTriggerBits(mask|AliAODForwardMult::kA)) nBg++;
884 if (mult->IsTriggerBits(mask|AliAODForwardMult::kC)) nBg++;
885 if (mult->IsTriggerBits(mask|AliAODForwardMult::kE)) nBg--;
56bd6baf 886
ffa07380 887 // Other trigger/event requirements could be defined
888 if (!mult->IsTriggerBits(mask)) continue;
fc6a90cc 889 nT++;
ffa07380 890
56bd6baf 891 // Check if we have vertex and select vertex range (in centimeters)
fc6a90cc 892 if (!mult->HasIpZ()) continue;
893 nV++;
895 if (!mult->InRange(vzLow, vzHigh) continue;
896 nA++;
ffa07380 897
898 // Add contribution from this event
899 sum->Add(&(mult->GetHistogram()));
900 }
655b45b0 901
ffa07380 902 // Get acceptance normalisation from underflow bins
fc6a90cc 903 TH1D* norm = sum->ProjectionX("norm", 0, 0, "");
ffa07380 904 // Project onto eta axis - _ignoring_underflow_bins_!
fc6a90cc 905 TH1D* dndeta = sum->ProjectionX("dndeta", 1, -1, "e");
ffa07380 906 // Normalize to the acceptance, and scale by the vertex efficiency
907 dndeta->Divide(norm);
fc6a90cc 908 dndeta->Scale(trigEff * nT/nV / (1 - nBg/nT), "width");
ffa07380 909 // And draw the result
910 dndeta->Draw();
0a89eed1 913
56bd6baf 914\section{$\Delta E$ fits}
918 \centering
dc64f2ea 919 \includegraphics[keepaspectratio,width=\textwidth]{eloss_fits}
920 \caption{Summary of energy loss fits in each $\eta$ bin (see also
921 \secref{sec:sub:sub:eloss_fits}).
922 \newline
923 On the left side: Top panel shows the
924 reduced $\chi^2$, second from the top shows the found
925 scaling constant, 3\textsuperscript{rd} from the top is
926 the most probable energy loss $\Delta_{mp}$, 4\textsuperscript{th}
927 shows the width parameter $\xi$ of the Landau, and the
928 5\textsuperscript{th} is the Gaussian width $\sigma$.
b9bd46b7 929 $\Delta_{mp}$, $\xi$, and $\sigma$ have units of $\Delta E/\Delta
930 E_{mip}$.
dc64f2ea 931 \newline
932 On the right: The top panel shows the maximum number of
933 multi--particle signals that where fitted, and the 4 bottom panels
934 shows the weights $a_2,a_3,a_4,$ and $a_5$ for 2, 3, 4, and 5
935 particle responses.}
56bd6baf 936 \label{fig:eloss_fits}
dc64f2ea 939\clearpage
0a89eed1 941\begin{thebibliography}{99}
56bd6baf 942\bibitem{FWD:2004mz} \ALICE{} Collaboration, Bearden, I.~G.\ \textit{et al}
943 \textit{ALICE technical design report on forward detectors: FMD, T0
944 and V0}, \CERN{}, 2004, CERN-LHCC-2004-025
945\bibitem{cholm:2009} Christensen, C.~H., \textit{The ALICE Forward
946 Multiplicity Detector --- From Design to Installation},
947 Ph.D.~thesis, University of Copenhagen, 2009,
948 \url{http://www.nbi.dk/~cholm/}.
dc64f2ea 949\bibitem{nim:b1:16}
950%% \bibitem{Hancock:1983ry}
951 S.~Hancock, F.~James, J.~Movchet {\it et al.},
952 ``Energy Loss Distributions For Single Particles And Several
953 Particles In A Thin Silicon Absorber,'' Nucl.\ Instrum.\ Meth.\
b9bd46b7 954 \textbf{B1} (1984) 16, \url{http://cdsweb.cern.ch/record/147286/files/cer-000058451.pdf}.
dc64f2ea 955\bibitem{phyrev:a28:615}
956%% \bibitem{Hancock:1983fp}
957 S.~Hancock, F.~James, J.~Movchet {\it et al.}, ``Energy Loss And
958 Energy Straggling Of Protons And Pions In The Momentum Range
b9bd46b7 959 0.7-gev/c To 115-gev/c,'' Phys.\ Rev.\ \textbf{A28} (1983) 615,
960 \url{http://cdsweb.cern.ch/record/145395/files/PhysRevA.28.615.pdf}.
0a89eed1 961\end{thebibliography}
655b45b0 962\end{document}
56bd6baf 963
964% Local Variables:
965% ispell-local-dictionary: "british"
966% End:
fc6a90cc 968% LocalWords: tracklet diffractive IsTriggerBits AliAODForwardMult ProjectionX