1 \documentclass[11pt]{article}
2 \renewcommand{\rmdefault}{ptm}
4 \usepackage[margin=2cm,twoside,a4paper]{geometry}
7 \usepackage[ruled,vlined,linesnumbered]{algorithm2e}
13 \usepackage[colorlinks,urlcolor=black,hyperindex,%
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17 %% \usepackage{bookmark}
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19 \newcommand{\AbbrName}[1]{\AlwaysText{{\scshape #1}}}
20 \newcommand{\CERN}{\AbbrName{cern}}
21 \newcommand{\ALICE}{\AbbrName{alice}}
22 \newcommand{\SPD}{\AbbrName{spd}}
23 \newcommand{\ESD}{\AbbrName{esd}}
24 \newcommand{\AOD}{\AbbrName{aod}}
25 \newcommand{\INEL}{\AbbrName{inel}}
26 \newcommand{\INELONE}{$\AbbrName{inel}>0$}
27 \newcommand{\NSD}{\AbbrName{nsd}}
28 \newcommand{\FMD}[1][]{\AbbrName{fmd\ifx|#1|\else#1\fi}}
29 \newcommand{\OCDB}{\AbbrName{ocdb}}
30 \newcommand{\mult}[1][]{\ensuremath N_{\text{ch}#1}}
31 \newcommand{\dndetadphi}[1][]{{\ensuremath%
32 \ifx|#1|\else\left.\fi%
33 \frac{d^2\mult{}}{d\eta\,d\varphi}%
34 \ifx|#1|\else\right|_{#1}\fi%
36 \newcommand{\landau}[1]{{\ensuremath%
37 \text{landau}\left(#1\right)}}
38 \newcommand{\dndeta}[1][]{{\ensuremath%
39 \ifx|#1|\else\left.\fi%
40 \frac{1}{N}\frac{d\mult{}}{d\eta}%
41 \ifx|#1|\else\right|_{#1}\fi%
43 \newcommand{\dndphi}[1][]{{\ensuremath%
44 \ifx|#1|\else\left.\fi%
45 \frac{1}{N}\frac{d\mult{}}{d\varphi}%
46 \ifx|#1|\else\right|_{#1}\fi%
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57 \newcommand{\cm}[1]{\unit[#1]{\AlwaysText{cm}}}
58 \newcommand{\secref}[1]{Section~\ref{#1}}
59 \newcommand{\figref}[1]{Figure~\ref{#1}}
60 \newcommand{\etaphi}{\ensuremath(\eta,\varphi)}
61 % Azimuthal acceptance
62 \newcommand{\Corners}{\ensuremath A^{\varphi}_{t}}
63 % Acceptance due to dead strips
64 \newcommand{\DeadCh}{\ensuremath A^{\eta}_{v,i}\etaphi}
65 \newcommand{\SecMap}{\ensuremath S_v\etaphi}
66 \setlength{\parskip}{1ex}
67 \setlength{\parindent}{0em}
69 {\LARGE EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH}\\%
70 {\Large European Organization for Particle Physics}\\[2ex]%
72 \begin{tabular}[t]{@{}p{.25\textwidth}@{}%
77 \includegraphics[keepaspectratio,width=.12\textwidth]{alice_logo_v3}%
82 {\LARGE\bf Analysing the FMD data for $\dndeta$}%
88 \begin{tabular}[t]{@{}p{.25\textwidth}@{}}
89 \hfill\includegraphics[keepaspectratio,width=.12\textwidth]{%
91 \hfill ALICE--INT--2012--040 v2\\
96 \author{Christian Holm
97 Christensen\thanks{\texttt{$\langle$cholm@nbi.dk$\rangle$}}\quad\&\quad
98 Hans Hjersing Dalsgaard\thanks{\texttt{$\langle$canute@nbi.dk$\rangle$}}\\
99 Niels Bohr Institute\\
100 University of Copenhagen}
103 \pdfbookmark{Analysing the FMD data for dN/deta}{top}
107 \section{Introduction}
109 This document describes the steps performed in the analysis of the
110 charged particle multiplicity in the forward pseudo--rapidity
111 regions with the \FMD{} detector \cite{FWD:2004mz,cholm:2009}. The
112 document also include a summary (see section \ref{prelim}) of the request for preliminary figures
113 for the measurement of $\dndeta$ with SPD\cite{ruben,Aamodt:2010cz},
114 VZERO\cite{maxime}, and FMD.
115 % The primary detector used for this is the \FMD{}
118 organised in 3 \emph{sub--detectors} \FMD{1}, \FMD{2}, and \FMD{3}, each
119 consisting of 1 (\FMD{1}) or 2 (\FMD{2} and~3) \emph{rings}.
120 The rings fall into two types: \emph{Inner} or \emph{outer} rings.
121 Each ring is in turn azimuthally divided into \emph{sectors}, and each
122 sector is radially divided into \emph{strips}. How many sectors,
123 strips, as well as the $\eta$ coverage is given in
124 \tablename~\ref{tab:fmd:overview}.
128 \caption{Physical dimensions of Si segments and strips.}
129 \label{tab:fmd:overview}
131 \begin{tabular}{|c|cc|cr@{\space--\space}l|r@{\space--\space}l|}
133 \textbf{Sub--detector/} &
137 \multicolumn{2}{c|}{\textbf{$r$}} &
138 \multicolumn{2}{c|}{\textbf{$\eta$}} \\
143 \multicolumn{2}{c|}{\textbf{range [cm]}} &
144 \multicolumn{2}{c|}{\textbf{coverage}} \\
146 FMD1i & 20& 512& 320 & 4.2& 17.2& 3.68& 5.03\\
147 FMD2i & 20& 512& 83.4& 4.2& 17.2& 2.28& 3.68\\
148 FMD2o & 40& 256& 75.2& 15.4& 28.4& 1.70& 2.29\\
149 FMD3i & 20& 512& -75.2& 4.2& 17.2&-2.29& -1.70\\
150 FMD3o & 40& 256& -83.4& 15.4& 28.4&-3.40& -2.01\\
156 The \FMD{} \ESD{} object contains the scaled energy deposited $\Delta
157 E/\Delta E_{mip}$ for each of the 51,200 strips. This is determined
158 in the reconstruction pass. The scaling to $\Delta E_{mip}$ is done
159 using calibration factors extracted in designated pulser runs. In
160 these runs, the front-end electronics is pulsed with an increasing
161 known pulse size, and the conversion factor from ADC counts to $\Delta
162 E_{mip}$ is determined \cite{cholm:2009}.
164 The \SPD{} is used for determination of the position of the primary
165 interaction point except in the case of displaced vertex analysis as
166 discussed in section \ref{sec:sub:sub:dispvtx}.
168 The analysis is performed as a two--step process.
170 \item The Event--Summary--Data (\ESD{}) is processed event--by--event
171 and passed through a number of algorithms, and
172 $\dndetadphi$ for each event is output to an Analysis--Object--Data
173 (\AOD{}) tree (see \secref{sec:gen_aod}).
174 \item The \AOD{} data is read in and the sub--sample of the data under
175 investigation is selected (e.g., \INEL{}, \INELONE{}, \NSD{} in p+p data, or
176 some centrality class in Pb+Pb data) and the $\dndetadphi$ histogram read for
177 those events to build up $\dndeta$ (see \secref{sec:ana_aod}).
179 The details of each step above will be expanded upon in the
182 In Appendix~\ref{app:nomen} is an overview of the nomenclature used in
187 \section{Generating $\dndetadphi[i]$ event--by--event}
190 When reading in the \ESD{}s and generating the $\dndetadphi$
191 event--by--event the following steps are taken (in order) for each
192 event $i$ and FMD ring $r$.
194 \item[Event inspection] The global properties of the event is
195 determined, including the trigger type and primary interaction
196 point\footnote{`Vertex' and `primary interaction point' will be used
197 interchangeably in the text, since there is no ambiguity with
198 particle production vertex in this analysis.} $z$ coordinate (see
199 \secref{sec:sub:event_inspection}).
200 \item[Sharing filter] The \ESD{} object is read in and corrected for
201 sharing. The result is a new \ESD{} object (see
202 \secref{sec:sub:sharing_filter}).
203 \item[Density calculator] The (possibly un--corrected) \ESD{} object
204 is then inspected and an inclusive (primary \emph{and} secondary
205 particles), per--ring charged particle density
206 $\dndetadphi[incl,r,v,i]$ is made. This calculation depends in
207 general upon the interaction vertex position along the $z$ axis
208 $v_z$ (see \secref{sec:sub:density_calculator}).
209 \item[Corrections] The 5 (one for each FMD ring) $\dndetadphi[incl,r,v,i]$ are corrected for
210 secondary production and acceptance. The correction for the
211 secondary particle production is highly dependent on the vertex $z$
212 coordinate. The result is a per--ring, charged primary particle
213 density $\dndetadphi[r,v,i]$ (see \secref{sec:sub:corrector}).
214 \item[Histogram collector] Finally, the 5 $\dndetadphi[r,v,i]$ are
215 summed into a single $\dndetadphi[v,i]$ histogram, taking care of
216 the overlaps between the detector rings. In principle, this
217 histogram is independent of the vertex, except that the
218 pseudo--rapidity range, and possible holes in that range, depends on
219 $v_z$ --- or rather the bin in which the $v_z$ falls (see
220 \secref{sec:sub:hist_collector}).
223 Each of these steps will be detailed in the following.
225 \subsection{Event inspection}
226 \label{sec:sub:event_inspection}
228 The first thing to do, is to inspect the event for triggers. A number
229 of trigger bits, like \INEL{} (Minimum Bias for Pb+Pb), \INELONE{}, \NSD{}, and so on is then
230 propagated to the \AOD{} output.
232 Just after the sharing filter (described below) but before any further
233 processing, the vertex information is queried. If there is no vertex
234 information, or if the vertex $z$ coordinate is outside the
235 pre--defined range, then no further processing of that event takes place.
237 \subsubsection{Displaced Vertices}
238 \label{sec:sub:sub:dispvtx}
240 The analysis can be set up to run on the `displaced vertices' that
241 occur during LHC Pb+Pb running. Details on the displaced vertices, and
242 their selection can be found in the VZERO analysis note \cite{maxime}.
243 \subsection{Sharing filter}
244 \label{sec:sub:sharing_filter}
246 A particle originating from the vertex can, because of its incident
247 angle on the \FMD{} sensors traverse more than one strip (see
248 \figref{fig:share_fraction}). This means that the energy loss of the
249 particle is distributed over 1 or more strips. The signal in each
250 strip should therefore possibly be merged with its neighboring strip
251 signals to properly reconstruct the energy loss of a single particle.
255 \includegraphics[keepaspectratio,height=3cm]{share_fraction}
256 \caption{A particle traversing 2 strips and depositing energy in
258 \label{fig:share_fraction}
261 The effect is most pronounced in low--flux\footnote{Events with a low
262 hit density.} events, like proton--proton collisions or peripheral
263 Pb--Pb collisions, while in high--flux events the hit density is so
264 high that most likely each and every strip will be hit and the effect
265 cancels out on average.
267 Since the particles travel more or less in straight lines toward the
268 \FMD{} sensors, the sharing effect is predominantly in the $r$ or
269 \emph{strip} direction. Only neighbouring strips in a given sector are
270 therefore investigated for this effect.
272 Algorithm~\ref{algo:sharing} is applied to the signals in a given
275 \begin{algorithm}[htpb]
276 \belowpdfbookmark{Algorithm 1}{algo:sharing}
277 \SetKwData{usedThis}{current strip used}
278 \SetKwData{usedPrev}{previous strip used}
279 \SetKwData{Output}{output}
280 \SetKwData{Input}{input}
281 \SetKwData{Nstr}{\# strips}
282 \SetKwData{Signal}{current}
283 \SetKwData{Eta}{$\eta$}
284 \SetKwData{prevE}{previous strip signal}
285 \SetKwData{nextE}{next strip signal}
286 \SetKwData{lowFlux}{low flux flag}
287 \SetKwFunction{SignalInStrip}{SignalInStrip}
288 \SetKwFunction{MultiplicityOfStrip}{MultiplicityOfStrip}
289 \usedThis $\leftarrow$ false\;
290 \usedPrev $\leftarrow$ false\;
291 \For{$t\leftarrow1$ \KwTo \Nstr}{
292 \Output${}_t\leftarrow 0$\;
293 \Signal $\leftarrow$ \SignalInStrip($t$)\;
295 \uIf{\Signal is not valid}{
296 \Output${}_t \leftarrow$ invalid\;
298 \uElseIf{\Signal is 0}{
299 \Output${}_t \leftarrow$ 0\;
302 \Eta$\leftarrow$ $\eta$ of \Input${}_t$\;
303 \prevE$\leftarrow$ 0\;
304 \nextE$\leftarrow$ 0\;
306 \prevE$\leftarrow$ \SignalInStrip($t-1$)\;
309 \nextE$\leftarrow$ \SignalInStrip($t+1$)\;
311 \Output${}_t\leftarrow$
312 \MultiplicityOfStrip(\Signal,\Eta,\prevE,\nextE,\\
313 \hfill\lowFlux,$t$,\usedPrev,\usedThis)\;
316 \caption{Sharing correction}
320 Here the function \FuncSty{SignalInStrip}($t$) returns the properly
321 path--length corrected signal in strip $t$. The function
322 \FuncSty{MultiplicityOfStrip} is where the real processing takes
323 place (see page \pageref{func:MultiplicityOfStrip}).
325 \begin{function}[htbp]
326 \belowpdfbookmark{MultiplicityOfStrip}{func:MultiplicityOfStrip}
327 \caption{MultiplicityOfStrip(\DataSty{current},$\eta$,\DataSty{previous},\DataSty{next},\DataSty{low
328 flux flag},\DataSty{previous signal used},\DataSty{this signal
330 \label{func:MultiplicityOfStrip}
331 \SetKwData{Current}{current}
332 \SetKwData{Next}{next}
333 \SetKwData{Previous}{previous}
334 \SetKwData{lowFlux}{low flux flag}
335 \SetKwData{usedPrev}{previous signal used}
336 \SetKwData{usedThis}{this signal used}
337 \SetKwData{lowCut}{low cut}
338 \SetKwData{total}{Total}
339 \SetKwData{highCut}{high cut}
340 \SetKwData{Eta}{$\eta$}
341 \SetKwFunction{GetHighCut}{GetHighCut}
342 \If{\Current is very large or \Current $<$ \lowCut} {
343 \usedThis $\leftarrow$ false\;
344 \usedPrev $\leftarrow$ false\;
348 \usedThis $\leftarrow$ false\;
349 \usedPrev $\leftarrow$ true\;
352 \highCut $\leftarrow$ \GetHighCut($t$,\Eta)\;
353 %\If{\Current $<$ \Next and \Next $>$ \highCut and \lowFlux set}{
354 % \usedThis $\leftarrow$ false\;
355 % \usedPrev $\leftarrow$ false\;
358 \total $\leftarrow$ \Current\;
359 \lIf{\lowCut $<$ \Previous $<$ \highCut and not \usedPrev}{
360 \total $\leftarrow$ \total + \Previous\;
362 \If{\lowCut $<$ \Next $<$ \highCut}{
363 \total $\leftarrow$ \total + \Next\;
364 \usedThis $\leftarrow$ true\;
367 \usedPrev $\leftarrow$ true\;
370 \usedPrev $\leftarrow$ false\;
371 \usedThis $\leftarrow$ false\;
375 Here, the function \FuncSty{GetHighCut} (see below) evaluates a fit to the energy
376 distribution in the specified $\eta$ bin (see also
377 \secref{sec:sub:density_calculator}). It returns
381 where $\Delta_{mp}$ is the most probable energy loss, and $w$ is the
382 width of the Landau distribution.
384 The \KwSty{if} in line 5, says that if the previous strip was merged
385 with current one, and the signal of the current strip was added to
386 that, then the current signal is set to 0, and we mark it as used for
387 the next iteration (\DataSty{previous signal used}$\leftarrow$true).
389 % The \KwSty{if} in line 10 checks if the current signal is smaller than
390 % the next signal, if the next signal is larger than the upper cut
391 % defined above, and if we have a low--flux event\footnote{Note, that in
392 % the current implementation there are never any low--flux events.}.
393 % If that condition is met, then the current signal is the smaller of
394 % two possible candidates for merging, and it should be merged into the
395 % next signal. Note, that this \emph{only} applies in low--flux events.
398 we test if the previous signal lies between our low and
399 high cuts, and if it has not been marked as being used. If so, we add
400 it to our current signal.
402 The next \KwSty{if} on line 12 % 16
403 checks if the next signal is within our
404 cut bounds. If so, we add that signal to the current signal and mark
405 it as used for the next iteration (\DataSty{this signal
406 used}$\leftarrow$true). It will then be put to zero on the next
407 iteration by the condition on line 6.
409 Finally, if our signal is still larger than 0, we return the signal
410 and mark this signal as used (\DataSty{previous signal
411 used}$\leftarrow$true) so that it will not be used in the next
412 iteration. Otherwise, we mark the current signal and the next signal
413 as unused and return a 0.
416 \subsection{Density calculator}
417 \label{sec:sub:density_calculator}
419 The density calculator loops over all the strip signals in the sharing
420 corrected\footnote{The sharing correction can be disabled, in which
421 case the density calculator uses the input \ESD{} signals.} \ESD{}
422 and calculates the inclusive (primary + secondary) charged particle
423 density in pre--defined $\etaphi$ bins.
425 \subsubsection{Inclusive number of charged particles: Energy Fits}
426 \label{sec:sub:sub:eloss_fits}
428 The number charged particles in a strip $\mult[,t]$ is calculated
429 using multiple Landau-like distributions fitted to the energy loss
430 spectrum of all strips in a given $\eta$ bin.
432 \Delta_{i,mp} &= i (\Delta_{1,mp}+ \xi_1 \log(i))\nonumber\\
433 \xi_i &= i\xi_1\nonumber\\
434 \sigma_i &= \sqrt{i}\sigma_1\nonumber\\
435 \mult[,t] &= \frac{\sum_i^{N_{max}}
436 i\,a_i\,F(\Delta_t;\Delta_{i,mp},\xi_i,\sigma_i)}{
437 \sum_i^{N_{max}}\,a_i\,F(\Delta_t;\Delta_{i,mp},\xi_i,\sigma_i)}\quad,
439 where $F(x;\Delta_{mp},\xi,\sigma)$ is the evaluation of the Landau
440 distribution $f_L$ with most probable value $\Delta_{mp}$ and width
441 $\xi$, folded with a Gaussian distribution with spread $\sigma$ at the
442 energy loss $x$ \cite{nim:b1:16,phyrev:a28:615}.
444 \label{eq:energy_response}
445 F(x;\Delta_{mp},\xi,\sigma) = \frac{1}{\sigma \sqrt{2 \pi}}
446 \int_{-\infty}^{+\infty} d\Delta' f_{L}(x;\Delta',\xi)
447 \exp{-\frac{(\Delta_{mp}-\Delta')^2}{2\sigma^2}}\quad,
449 where $\Delta_{1,mp}$, $\xi_1$, and $\sigma_1$ are the parameters for
450 the first MIP peak, $a_1=1$, and $a_i$ is the relative weight of the
451 $i$-fold MIP peak. The parameters $\Delta_{1,mp}, \xi_1,
452 \sigma_1, \mathbf{a} = \left(a_2, \ldots a_{N_{max}}\right)$ are
455 F_j(x;C,\Delta_{mp},\xi,\sigma,\mathbf{a}) = C
456 \sum_{i=1}^{j} a_i F(x;\Delta_{i,mp},\xi_{i},\sigma_i)
458 for increasing $j$ to the energy loss spectra in separate $\eta$ bins.
459 The fit procedure is stopped when for $j+1$: (the default values for
460 each value are included below)
462 \item the reduced $\chi^2$ exceeds a certain threshold (usually 20), or
463 \item the relative error $\delta p/p$ of any parameter of the fit
464 exceeds a certain threshold (usually 0.12), or
465 \item when the weight $a_j+1$ is smaller than some number (typically
468 $N_{max}$ is then set to $j$. Examples of the result of these fits
469 are given in \figref{fig:eloss_fits} in Appendix~\ref{app:eloss_fits}.
470 \subsubsection{Inclusive number of charged particles: Poisson Approach}
471 \label{sec:sub:sub:poisson}
472 Another approach to the calculation of the number of charged particles
473 is using Poisson statistics. This is the default choice because it is
474 less sensitive to the stability of the fits required for the energy
476 Assume that in a region of the FMD % where
478 %is azimuthally uniform in $\eta$ intervals it
480 distributed according to a Poisson distribution. This means that the
481 probability of $\mult=n$ becomes:
483 P(n) = \frac{\mu^n e^{-\mu}}{n!} \label{eq:PoissonDef}
485 In particular the measured occupancy, $\mu_{meas}$, is the probability
486 of any number of hits, thus using \eqref{eq:PoissonDef} :
488 \mu_{meas} = 1 - P(0) = 1 - e^{-\mu }
489 %\Rightarrow \mu = \ln
490 %(1 - \mu_{meas})^{-1} \label{eq:PoissonDef2}
495 (1 - \mu_{meas})^{-1} \label{eq:PoissonDef2}
497 The mean number of particles in a hit strip becomes:
499 C &=& \frac{\sum_{n>0} n P(n>0)}{\sum_{n>0} P(n>0)} \nonumber \\
500 &=& \frac{e^{-\mu}}{1-e^{-\mu}} \mu \sum \frac{\mu^n}{n!}
502 &=& \frac{e^{-\mu}}{1-e^{-\mu}} \mu e^{\mu} \nonumber \\
503 &=& \frac{\mu}{1-e^{-\mu}}
505 %While $\mu$ can be calculated analytically for practical purposes we
506 With $\mu$ defined in \eqref{eq:PoissonDef2} this calculation is
507 carried out per event in
508 regions of the FMD each containing 256 strips\footnote{Note that this means that the same factor is used for each of the 256 strips.}. %Defining
509 %$\mu_{meas}^{region}$ to be the measured occupancy
511 $\mult$ for a hit strip ($N_{hits} \equiv 1$) in that region becomes:
513 \mult = N_{hits} \times C = 1 \times C = C
515 Where C is calculated using $\mu_{meas}^{region}$.
517 The Poisson method and the energy fits method have been compared in
518 \cite{hhd:2009} where it is found that the two methods are in good
519 agreement. The residual difference between the methods contributes to
520 the systematic error.
522 \subsubsection{Azimuthal Acceptance}
524 Before the signal $\mult[,t]$ can be added to the $\etaphi$
525 bin in one of the 5 per--ring histograms, it needs to be corrected for
526 the $\varphi$ acceptance of the strip.
528 The sensors of the \FMD{} are not perfect arc--segments --- the two
529 top corners are cut off to allow the largest possible sensor on a 6''
530 Si-wafer. This means, however, that the strips in these outer
531 regions do not fully cover $2\pi$ in azimuth, and there is therefore a
532 need to correct for this limited acceptance.
534 The acceptance correction is only applicable where the strip length
535 does not cover the full sector. This is the case for the outer strips
536 in both the inner and outer type rings. The acceptance correction is
540 \Corners{} &= \frac{l_t}{\Delta\varphi}\quad
542 where $l_t$ is the strip length in radians at constant $r$, and
543 $\Delta\varphi$ is $2\pi$ divided by the number of sectors in the
544 ring (20 for inner type rings, and 40 for outer type rings).
546 Note, that this correction is a hardware--related correction, and does
547 not depend on the properties of the collision (e.g., primary vertex
550 The final $\etaphi$ content of the 5 output vertex dependent,
551 per--ring histograms of the inclusive charged particle density is then
555 \dndetadphi[incl,r,v,i\etaphi] &= \sum_t^{t\in\etaphi}
556 \mult[,t]\,\Corners{}
558 where $t$ runs over the strips in the $\etaphi$ bin.
560 \subsection{Corrections}
561 \label{sec:sub:corrector}
563 The corrections code receives the five vertex dependent,
564 per--ring histograms of the inclusive charged particle density
565 $\dndetadphi[incl,r,v,i]$ from the density calculator and applies
568 \subsubsection{Secondary correction}
570 %% hHits_FMD<d><r>_vtx<v>
571 %% hCorrection = -----------------------
572 %% hPrimary_FMD_<r>_vtx<v>
575 %% - hPrimary_FMD_<r>_vtx<vtx> is 2D of eta,phi for all primary ch
577 %% - hHits_FMD<d><r>_vtx<v> is 2D of eta,phi for all track-refs that
578 %% hit the FMD - The 2D version of hMCHits_nocuts_FMD<d><r>_vtx<v>
580 This is a 2 dimensional histogram generated from simulations, as the
581 ratio of primary particles to the total number of particles that fall
582 within an $\etaphi$ bin for a given vertex bin
587 \frac{\sum_i^{\NV[,v]}\mult[,\text{primary},i]\etaphi}{
588 \sum_i^{\NV[,v]}\mult[,\text{\FMD{}},i]\etaphi}\quad,
590 where $\NV[,v]$ is the number of events with a valid trigger and a
591 vertex in bin $v$, and $\mult[,\FMD{},i]$ is the total number of
592 charged particles that hit the \FMD{} in event $i$ in the specified
593 $\etaphi$ bin and $\mult[,\text{primary},i]$ is number of
594 primary charged particles in event $i$ within the specified
597 $\mult[,\text{primary}]\etaphi$ is given by summing over the
598 charged particles labelled as primaries \emph{at the time of the
599 collision} as defined in the simulation code. That is, it is the
600 number of primaries within the $\etaphi$ bin at the collision
601 point --- not at the \FMD{}.
603 $\SecMap$ varies from $\approx 1.5$ for the most forward bins to
604 $\approx 3$ for the more central bins. Figure \ref{secondaries} shows the $\dndeta$ of secondaries from various sources assessed with MC simulations to give an idea of the magnitude of the effects of secondaries.
607 \includegraphics[keepaspectratio,width=\textwidth]{%
609 \caption{$\dndeta$ for secondaries and primaries in the FMD. The same plot for the SPD inner layer is included for comparison.}
613 %For pp, different event
614 %generators were used and found to give compatible results within
616 For pp, at least some millions of events must be
617 accumulated to reach satisfactory statistics. For Pb--Pb where the
618 general hit density is larger, reasonable statistics can be achieved
619 with less simulated data.
621 \subsubsection{Acceptance due to dead channels}
623 Some of the strips in the \FMD{} have been marked up as \emph{dead},
624 meaning that they are not used in the reconstruction or analysis.
625 This leaves holes in the acceptance of each defined $\etaphi$
626 which need to be corrected for.
628 Dead channels are marked specially in the \ESD{}s with the flag
629 \textit{Invalid Multiplicity}. This is used in the analysis to build
630 up and event--by--event acceptance correction in each $\etaphi$
631 bin by calculating the ratio
633 \label{eq:dead_channels}
635 \frac{\sum_t^{t\in\etaphi}\left\{\begin{array}{cl}
636 1 & \text{if not dead}\\
638 \end{array}\right.}{\sum_t^{t\in\etaphi} 1}\quad,
640 where $t$ runs over the strips in the $\etaphi$ bin. This correction
641 is obviously $v_z$ dependent since the $\etaphi$ bin to which a strip $t$
642 corresponds to depends on its position relative to the primary vertex.
644 Alternatively, pre--made acceptance factors can be used. These are
645 made from the off-line conditions database (\OCDB{}).
647 The 5 output vertex dependent, per--ring histograms of the primary
648 charged particle density is then given by
650 \dndetadphi[r,v,i\etaphi] &=
651 \SecMap{} \frac{1}{\DeadCh{}}\dndetadphi[incl,r,v,i\etaphi]
654 \subsection{Histogram collector}
655 \label{sec:sub:hist_collector}
657 The histogram collector collects the information from the 5 vertex
658 dependent, per--ring histograms of the primary charged particle
659 density $\dndetadphi[r,v,i]$ into a single vertex dependent histogram
660 of the charged particle density $\dndetadphi[v,i]$.
662 To do this, it first calculates, for each vertex bin, the $\eta$ bin
663 range to use for each ring. It investigates the secondary correction
664 maps $\SecMap{}$ to find the edges of each map. The edges are given
665 by the $\eta$ range where $\SecMap{}$ is larger than some
666 threshold\footnote{Typically $t_s\approx 0.1$.} $t_s$. The code
667 applies safety margin of a number of bins, $N_{cut}$\footnote{Typically
668 $N_{cut}=1$.}, to ensure that the data selected does not have too
669 large corrections associated with it.
671 It then loops over the bins in the defined $\eta$ range and sums the
672 contributions from each of the 5 histograms. In the $\eta$ ranges
673 where two rings overlap, the collector calculates the average and adds
674 the errors in quadrature\footnote{While not explicitly checked, it was
675 found that the histograms agrees within error bars in the
678 The output vertex dependent histogram of the primary
679 charged particle density is then given by
682 \dndetadphi[v,i\etaphi] &=
683 \frac{1}{N_{r\in\etaphi}}\sum_{r}^{r\in\etaphi}
684 \dndetadphi[r,v,i\etaphi]\\
685 \delta\left[\dndetadphi[v,i\etaphi]\right] &=
686 \frac{1}{N_{r\in\etaphi}}\sqrt{\sum_{r}^{r\in\etaphi}
687 \delta\left[\dndetadphi[r,v,i\etaphi]\right]^2}
690 where $N_{r\in\etaphi}$ is the number of overlapping histograms
691 in the given $\etaphi$ bin.
693 The histogram collector stores the found $\eta$ ranges in the
694 underflow bin of the histogram produced. The content of the overflow
699 \frac{1}{N_{r\in(\eta)}}
700 \sum_{r}^{r\in(\eta)} \left\{\begin{array}{cl}
701 0 & \eta \text{\ bin not selected}\\
702 1 & \eta \text{\ bin selected}
703 \end{array}\right.\quad,
705 where $N_{r\in(\eta)}$ is the number of overlapping histograms in the
706 given $\eta$ bin. The subscript $v$ indicates that the content
707 depends on the current vertex bin of event $i$.
709 \section{Building the final $\dndeta$}
712 To build the final $\dndeta$ distribution it is enough to sum
713 \eqref{eq:superhist} and \eqref{eq:overflow} over all accepted
714 events, $\NA$, and correct for the acceptance $I(\eta)$
716 \dndetadphi[\etaphi] &= \sum_i^{\NA}\dndetadphi[i,v\etaphi]\\
717 I(\eta) &= \sum_i^{\NA}I_{i,v}(\eta)\quad.
719 Note, that $I(\eta)\le\NA$.
721 We then need to normalise to the total number of events $N_X$, given
724 \N{X}{} &= \frac{1}{\epsilon_X}\left[\NA + \alpha(\NnotV -
725 \beta)\right] \label{eq:fulleventnorm}\\
726 & = \frac{1}{\epsilon_X}\left[\NA + \frac{\NA}{\NV}(\NT-\NV{} -
727 \beta)\right]\nonumber \\
728 & =\frac{1}{\epsilon_X}\NA\left[1+\frac{1}{\epsilon_V}-1-
729 \frac{\beta}{\NV}\right]\nonumber\\
730 & = \frac{1}{\epsilon_X}\frac{1}{\epsilon_V}\NA
731 \left(1-\frac{\beta}{\NT{}}\right)\nonumber
735 \item[$\epsilon_X$] is the trigger efficiency for type
736 $X\in[\text{\INEL},\text{\INELONE},\text{\NSD} for p+p data and MB
738 \item[$\epsilon_V=\frac{\NV{}}{\NT{}}$] is the vertex efficiency
739 evaluated over the data.
740 \item[$\NA$] is the number of events with a trigger \emph{and} a valid
741 vertex in the selected range
742 \item[$\NV{}$] is the number of events with a trigger \emph{and} a valid
744 \item[$\NT$] is the number of events with a trigger.
745 \item[$\NnotV{}=\NT-\NV{}$] is the number of events with a trigger
746 \emph{but no} valid vertex
747 \item[$\alpha=\frac{\NA}{\NV}$] is the fraction of accepted events of
748 the total number of events with a trigger and valid vertex.
749 \item[$\beta=\N{a}{}+\N{c}{}-\N{e}{}$] is the number of background
750 events \emph{with} a valid off-line trigger. This formula is the
751 simplest case of one bunch crossing per trigger/background
752 class. For more complicated collision setups the fractions in the
755 The two terms under the parenthesis in \eqref{eq:fulleventnorm} refers
756 to the observed number of event $\NA$, and the events missed because
757 of no vertex reconstruction. Note, for $\beta\ll\NT{}$
758 \eqref{eq:fulleventnorm} reduces to the simpler expression
760 \N{X}{} = \frac1{\epsilon_X}\frac1{\epsilon_V}\NA{}
762 The trigger efficiency $\epsilon_X$ for a given trigger type $X$ is
763 evaluated from simulations as
765 \epsilon_X = \frac{\N{X\wedge \text{T}}{}}{\N{X}{}}\quad,
767 that is, the ratio of number of events of type $X$ with a
768 corresponding trigger to the number of events of type $X$.
770 The final event--normalised charged particle density then becomes
772 \frac{1}{N}\frac{dN_{\text{ch}}}{d\eta} &=
773 \frac{1}{\N{X}{}} \int_0^{2\pi} d\varphi
774 \frac{\dndetadphi[\etaphi]}{I(\eta)}
775 \label{eq:eventnormdndeta}
778 If the trigger $X$ introduces a bias on the measured number of events,
779 then \eqref{eq:eventnormdndeta} need to be modified to
781 \frac{1}{N}\frac{dN_{\text{ch}}}{d\eta} &=
782 \frac{1}{\N{X}{}} \int_0^{2\pi} d\varphi
783 \frac{\frac{1}{B\etaphi}\dndetadphi[\etaphi]}{I(\eta)}
784 \label{eq:eventnormdndeta2}\quad,
786 where $B\etaphi$ is the bias correction. This is typically
787 calculated from simulations using the expression
789 B\etaphi = \frac{\frac{1}{\N{X\wedge
790 \text{T}}{}}\sum_i^{\N{X\wedge \text{T}}{}}
791 \mult[,\text{primary}]\etaphi}{\frac{1}{\N{X}{}}\sum_i^{\N{X}{}}
792 \mult[,\text{primary}]\etaphi}
795 \section{Systematic Errors} \label{fmdsysterror}
798 \begin{tabular}{|c|c|c|}
800 Effect & Magnitude in Pb+Pb analysis & Magnitude in p+p
803 Variation of the cuts in sec. \ref{sec:sub:sharing_filter} & 2\% & 3\% \\
805 Calculation of $\mult$ & 3\% & 4\% \\
807 Material budget & 7 \% & 7 \% \\
809 Generator & 2\% & 2\% \\
811 Vertex and trigger bias & N/A & 3\% \\
813 Centrality & 1\% --6\% & N/A \\
815 Normalization & N/A & 1.3\% - 3\% \\
818 Total in quadrature & 8.2\% -- 10.1\% & 9.4 \% -- 9.8\% \\
821 \caption[Systematic Errors in the FMD]{The table summarizes the
822 systematic errors in the FMD including the total systematic error
823 obtained by addition in quadrature.} \label{systerrors}
825 The systematic errors on the $\dndeta$ measurement are discussed in detail in
826 \cite{hhd:2009}. The results for the systematic errors in p+p and
827 Pb+Pb data are shown in table \ref{systerrors}. A short summary of the elements of the table is given here:
829 \item The variations of the cuts in section \ref{sec:sub:sharing_filter} are carried out by re--running the analysis with different cuts and taking the observed differences as the contribution to the systematic error.
830 \item To assess the error on the calculation of the multiplcity the two methods for counting particles (see section \ref{sec:sub:density_calculator}) are compared.
831 \item The systematic error on the material budget description was found from simulations with $\pm 10 \%$ increased density.
832 \item Several event generators were used to assess the error from the particular choice of generator in the analysis. The same procedure was used to assess the error from the MC dependent part of the correction for trigger and vertex bias (p+p only).
833 \item The systematic error on the centrality selection was obtained from variations of the different methods for measuring centrality.
836 \section{Using the per--event $\dndetadphi[i,v]$ histogram for other
839 \subsection{Multiplicity distribution}
841 To build the multiplicity distribution for a given $\eta$ range
842 $[\eta_1,\eta_2]$, one needs to find the total multiplicity in that
843 $\eta$ range for each event. To do so, one should sum the
844 $\dndetadphi[i,v]$ histogram over all $\varphi$ and in the selected
847 n'_{i[\eta_1,\eta_2]}, &= \int_{\eta_1}^{\eta_2}d\eta\int_0^{2\pi}d\varphi
848 \dndetadphi[i,v]\quad.\nonumber
850 However, $n'_i$ is not corrected for the coverage in $\eta$ for the
851 particular vertex range $v$. One therefor needs to correct for the
852 number of missing bins in the range $[\eta_1,\eta_2]$. Suppose
853 $[\eta_1,\eta_2]$ covers $N_{[\eta_1,\eta_2]}$ $\eta$ bins, then the acceptance
854 correction is given by
856 A_{i,[\eta_1,\eta_2]} = \frac{N_{[\eta_1,\eta_2]}}{\int_{\eta_1}^{\eta_2}d\eta\,
857 I_{i,v}(\eta)}\quad.\nonumber
859 The per--event multiplicity is then given by
861 n_{i,[\eta_1,\eta_2]} &= A_{i,[\eta_1,\eta_2]}\,n'_{i,[\eta_1,\eta_2]}\nonumber\\
862 &= \frac{N_{[\eta_1,\eta_2]}}{\int_{\eta_1}^{\eta_2}\eta
863 I_{i,v}(\eta)} \int_{\eta_1}^{\eta_2}d\eta\int_0^{2\pi}d\varphi
868 \subsection{Forward--Backward correlations}
870 To do forward--backward correlations, one need to calculate
871 $n_{i,[\eta_1,\eta_2]}$ as shown in \eqref{eq:event_n} in two bins
872 $n_{i,[\eta_1,\eta_2]}$ and $n_{i,[-\eta_2,-\eta_1]}$ \textit{e.g.},
873 $n_{i,f}=n_{i,[-3,-1]}$ and $n_{i,b}=n_{i,[1,3]}$.
876 \section{Some results}
878 %% \figurename{}s \ref{fig:1} to \ref{fig:3} shows some results.
879 Figures below show some examples \cite{hhd:2009}. Note these are not finalised
883 \includegraphics[keepaspectratio,width=\textwidth]{%
885 \caption{$\dndeta$ for pp for \INEL{} events at
886 $\sqrt{s}=\GeV{900}$, $\sqrt{s}=\TeV{2.76}$, and $\sqrt{s}=\TeV{7}$
887 $\cm{-10}\le v_z\le\cm{10}$, rebinned by a factor 5 \cite{hhd:2009}.
889 % shows the ratio of ALICE data to UA5, and the bottom panel shows
890 % the ratio of the right (positive) side to the left (negative) side
891 % of the forward $\dndeta$.
897 \includegraphics[keepaspectratio,width=\textwidth]{%
899 \caption{$\dndeta$ for Pb+Pb for Minimum Bias events at
900 $\sqrt{s_{NN}}=\TeV{2.76}$ $\cm{-10}\le v_z\le\cm{10}$, rebinned by a
901 factor 5 in 10 centrality intervals \cite{hhd:2009}.
903 % shows the ratio of ALICE data to UA5, and the bottom panel shows
904 % the ratio of the right (positive) side to the left (negative) side
905 % of the forward $\dndeta$.
914 \includegraphics[keepaspectratio,width=\textwidth]{%
915 dndeta_pp_0900GeV_INEL_m10p10cm}
916 \caption{$\dndeta$ for pp for \INEL{} events at $\sqrt{s}=\GeV{900}$,
917 $\cm{-10}\le v_z\le\cm{10}$, rebinned by a factor 5. Middle panel
918 shows the ratio of ALICE data to UA5, and the bottom panel shows
919 the ratio of the right (positive) side to the left (negative) side
920 of the forward $\dndeta$.}
927 \includegraphics[keepaspectratio,width=\textwidth]{%
928 dndeta_0900GeV_m10-p10cm_rb05_inelgt0}
929 \caption{$\dndeta$ for pp for \INELONE{} events at
930 $\sqrt{s}=\GeV{900}$, $\cm{-10}\le v_z\le\cm{10}$, rebinned by a
931 factor 5. Comparisons to other measurements shown where
937 \includegraphics[keepaspectratio,width=\textwidth]{%
938 dndeta_0900GeV_m10-p10cm_rb05_nsd}
939 \caption{$\dndeta$ for pp for \NSD{} events at $\sqrt{s}=\GeV{900}$,
940 $\cm{-10}\le v_z\le\cm{10}$, rebinned by a factor 5. Comparisons
941 to other measurements shown where applicable}
946 \section{Analysis for QM 2012 and Paper} \label{prelim}
947 \subsection{Analysis}
948 Following the development of the displaced vertex technique for VZERO \cite{maxime} it was
949 decided also to attempt such an analysis with the FMD using exactly
950 the same event selection and centrality selection as the VZERO
953 The analysis described in this note was used successfully
954 on these special events. Three detectors contribute to this
955 measurement: SPD with tracklets covering $-2<\eta<2$ \cite{ruben,Aamodt:2010cz}, VZERO covering
956 $-3<\eta<-1.25$ and $1.25<\eta<5.25$, and FMD covering $-5<\eta<-1.25$
957 and $1.25<\eta<5.5$. The extended coverage of the VZERO and FMD comes
958 from the positions of the displaced vertices. The full pseudorapidity
959 coverage of the combined measurement is $-5<\eta<5.5$.
961 To combine the measurements the individual measurements were weighted by
962 their systematic error before a weighted average was taken to form the
963 final $\dndeta$. The systematic error is calculated as an average in
964 quadrature with a contribution from the residual difference between
967 Due to the nature of the ZDCvsZEM centrality determination (see
968 \cite{maxime} for details) the centrality selection of the measurement
969 with SPD, VZERO, and FMD is limited to $30\%$ central collisions. The
970 centrality bins considered are thus $0-5\%$, $5-10\%$, $10-20\%$, and
973 The selected vertices with full pseudorapidity coverage for FMD in
974 this analysis are $\cm{112.5}$, $\cm{150}$, $\cm{187.5}$, $\cm{225}$,
975 $\cm{262.5}$, $\cm{300}$. For vertices $v_z > \cm{300}$ and $v_z <
976 \cm{112.5}$ a cut is imposed in pseudorapidity to only accept data
977 with $|\eta| > 4$ to avoid regions in ALICE known to have issues with
978 the material budget description.
980 \subsection{Analysis Performance}
981 This section includes some plots to assess the validity of the
982 analysis. This includes comparisons between the measurements used
983 (SPD, VZERO, and FMD) and
984 $\dndphi$ from the FMD.
986 Figure \ref{coverage} shows the pseudorapidity coverage of the FMD when using FMD1
987 and FMD2I as a function of vertex with displaced vertices.
990 \includegraphics[keepaspectratio,width=\textwidth]{coverage}
991 \caption{Pseudorapidity coverage of the FMD as a function of vertex
992 with displaced vertices.}
996 Figure \ref{spdfmdvzero} shows the results of the measurements of the
997 SPD, VZERO, and FMD. It is
998 seen that there is good
999 agreement between the three different measurements albeit residual
1000 differences of up to $6 \%$ remain.
1003 \includegraphics[keepaspectratio,width=\textwidth]{spdfmdvzero}
1004 \caption{$\dndeta$ measured with nominal vertices with the SPD and
1005 displaced vertices with VZERO and FMD. It is seen that there is
1006 good agreement between the measurements.}
1010 Figure \ref{ratiofmdvzero} shows the ratios of the measurements of FMD and
1011 VZERO to the combined measurement and to the SPD measurement. It is
1012 seen that the residual differences are small and there is good
1013 agreement between the three different measurements.
1016 \begin{minipage}{0.5\linewidth}
1018 \includegraphics[keepaspectratio,width=\textwidth]{ratiofmdvzero}
1020 \begin{minipage}{0.5\linewidth}
1022 \includegraphics[keepaspectratio,width=\textwidth]{ratiospdfmdvzero}
1024 \caption{Left: Ratio of FMD and VZERO measurements to the combined
1025 $\dndeta$ measured with SPD, VZERO and FMD. Right: Ratios of FMD
1026 and VZERO measurement to SPD measurement in regions of
1027 overlap. It is worth pointing out that the residual differences
1028 can come from the fact that the VZERO analysis uses SPD for
1029 absolute calibration while the FMD analysis does not. This means that the
1030 centrality determination for displaced vertices will affect the
1031 FMD analysis the most because the VZERO analysis has an additional
1032 constraint from the SPD analysis that uses the ZDCvsZEM centrality
1033 at midrapidity where it can be crosschecked with other means of
1034 centrality determination. Such a crosscheck is not possible elsewhere.}
1035 \label{ratiofmdvzero}
1038 Since $\dndeta$ is an average taken over $\varphi$ it is instructive to
1039 consider $\dndphi$ to check that these distributions are flat as they
1040 should be. Figure \ref{dndphi_pos} shows examples of the $\dndphi$
1041 distributions for FMD1. Figure \ref{dndphi_neg} shows examples from
1042 FMD2 (inner ring). The two low points at $\varphi \sim 5.5$ in
1043 Figure \ref{dndphi_neg} are
1044 understood as coming from two dying chips in FMD2I. They are considered dead
1045 in the final analysis and corrected for. It is seen that the trends
1046 are quite flat within $\sim 5\%$
1047 as expected. The same trend is observed for all the distributions.
1050 \includegraphics[keepaspectratio,width=\textwidth]{dNdphi040612}
1051 \caption{Examples of $\dndphi$ from FMD1 (positive
1052 pseudorapidities). The distributions are essentially flat.}
1057 \includegraphics[keepaspectratio,width=\textwidth]{dNdphi_neg_040612}
1058 \caption{Examples of $\dndphi$ from FMD2I (negative
1059 pseudorapidities). The two low points at $\varphi \sim 5.5$ are
1060 understood as the result of two dying chips in FMD2I. They are considered dead
1061 in the final analysis and corrected for accordingly. Apart from
1062 these points, the distributions are essentially flat.}
1065 Figure \ref{pervertex} shows the analysis performed for each
1066 vertex. The material budget effects for vertices $<\cm{112.5}$ are
1071 %\begin{minipage}{\linewidth}
1072 %\begin{minipage}{\columnwidth}
1074 \includegraphics[keepaspectratio,width=0.8\textwidth]{dNdeta_per_vertex160612_negfield}
1076 % \begin{minipage}{\linewidth}
1077 %\begin{minipage}{\columnwidth}
1079 \includegraphics[keepaspectratio,width=0.8\textwidth]{dNdeta_per_vertex160612_posfield}
1081 \caption{Top: Analysis per vertex for negative field data. Bottom:
1082 Analysis per vertex for positive field data. In the two plots the
1083 vertices where the full coverage is used are shown in blue. For the
1084 red and green points there a cut is applied for the pseudorapidity
1085 so that only points with $|\eta|>4$ are used in the analysis.}
1088 Figure \ref{leftright} shows the ratio of the postive and negative pseudorapidities for the FMD. It is seen that there are discrepancies of up to $\sim 5 \%$.
1091 \includegraphics[keepaspectratio,width=0.7\textwidth]{disp_dndeta_ratios_leftright}
1092 \caption{Ratios of the positive and negative pseudorapidities for the FMD (ratio is negative over positive). The grey band indicates the combined systematic error for FMD1I and FMD2I assuming excluding all contributions from event selection and material budget (i.e. the minimum systematic error between FMD1I and FMD2I).} \label{leftright}
1095 \subsection{Results}
1096 This section summarizes the final results of the analysis and includes
1097 the figures for approval.
1099 Figure \ref{combineddndeta} shows the combined $\dndeta$ from SPD,
1100 VZERO, and FMD in the full pseudorapidity range of $-5<\eta<5.5$.
1103 \includegraphics[keepaspectratio,width=\textwidth]{combineddndeta}
1104 \caption{Request for ALICE preliminary: Combined $\dndeta$ measured with SPD, VZERO and FMD. The
1105 VZERO and FMD measurements are made with displaced vertices and
1106 the SPD measurement is made at the nominal vertex. The fits are
1107 fits to a function $f(\eta) = A\exp (\frac{\eta -a_1}{2 a_2^2}) -
1108 B\exp (\frac{\eta -b_1}{2 b_{2}^2})$ i.e. a Gaussian centered on
1109 $ \eta = 0$ subtracted from a similar Gaussian.}
1110 \label{combineddndeta}
1113 Figure \ref{dndetaoverNpart} shows $dN/d\eta/(N_{part}/2)$ based on
1114 figure \ref{combineddndeta} and data taken from \cite{Aamodt:2010cz}.
1117 \includegraphics[keepaspectratio,width=\textwidth]{dndetaoverNpart}
1118 \caption{Request for ALICE preliminary: The $dN/d\eta/(N_{part}/2)$ measured with SPD, VZERO and FMD. The
1119 VZERO and FMD measurements are made with displaced vertices and
1120 the SPD measurement is made at the nominal vertex. The values of
1121 $N_{part}$ and the measurement at $-0.5<\eta<0.5$ taken from \cite{Aamodt:2010cz}.}
1122 \label{dndetaoverNpart}
1124 Using figure \ref{dndetaoverNpart}, figure \ref{RatiodndetaoverNpart}
1125 is constructed. It shows the ratios of $dN/d\eta/(N_{part}/2)$ in the
1126 following $\eta$ bins:
1127 $0.5<\eta<1.5$, $1.5<\eta<2.5$, $2.5<\eta<3.5$, $3.5<\eta<4.5$, and
1128 $4.5<\eta<5.5$ to the published $dN/d\eta/(N_{part}/2)$ at $-0.5<\eta<0.5$. These
1129 ratios are found to be flat for all pseudorapidity intervals.
1132 \includegraphics[keepaspectratio,width=\textwidth]{RatiodndetaoverNpart}
1133 \caption{Request for ALICE preliminary: Ratios of $dN/d\eta/(N_{part}/2)$ at
1134 $0.5<\eta<1.5$, $1.5<\eta<2.5$, $2.5<\eta<3.5$, $3.5<\eta<4.5$, and
1135 $4.5<\eta<5.5$ to the published $dN/d\eta/(N_{part}/2)$ at $-0.5<\eta<0.5$. The ratios are found to be flat for all the pseudorapidity intervals.}
1136 \label{RatiodndetaoverNpart}
1138 With the analysis presented in figure \ref{combineddndeta} it is also
1139 possible to study longitudinal scaling from LHC to RHIC
1140 energies. Figure \ref{longscaling} shows $\dndeta$ as a function of
1141 $y'=\eta-y_{beam}$ from Figure \ref{combineddndeta} and results from
1142 the BRAHMS\cite{Bearden:2001qq} and PHOBOS\cite{Alver:2010ck}
1143 experiments at RHIC. From the figure it is seen
1144 that with the wide coverage of the SPD, VZERO, and FMD measurement it
1145 is indeed likely that longitudinal scaling exist from RHIC to LHC
1149 \includegraphics[keepaspectratio,width=\textwidth]{longscaling}
1150 \caption{Request for ALICE preliminary: Study of Longitudinal
1151 scaling. $\dndeta$ as a function of
1152 $y'=\eta-y_{beam}$ from Figure \ref{combineddndeta} and the BRAHMS\cite{Bearden:2001qq} and
1153 PHOBOS\cite{Alver:2010ck} experiments at RHIC. The fits are the function
1154 from figure \ref{combineddndeta} and a straight line ending in
1155 $\eta=y_{beam}$. From the figure it seems likely that
1156 longitudinal scaling exists from RHIC to LHC energies.}
1159 Finally the total number of produced charged particles,
1160 $N_{ch}=\int^{y_{beam}}_{-y_{beam}}\dndeta d\eta$, has
1161 been calculated from the fits in Figure \ref{combineddndeta}. The
1162 obtained values of $N_{ch}$ versus $N_{part}$ are shown in figure
1163 \ref{totalNch}. The systematic errors on $N_{ch}$ have been assessed
1164 by the procedure of varying fit functions discussed in \cite{maxime}.
1167 \includegraphics[keepaspectratio,width=\textwidth]{totalNch}
1168 \caption{Request for ALICE preliminary: Total number of charged
1169 particles, $N_{ch}=\int^{y_{beam}}_{-y_{beam}}\dndeta d\eta$,
1170 obtained from the fitted function in figure
1171 \ref{combineddndeta}. The systematic errors on this plot were
1172 assessed by variation of the fit function as described in \cite{maxime}.}
1175 \subsection{Comparison to old Preliminary}
1176 At QM 2011 figures were approved for preliminary status and
1177 shown. Roughly six months later it was found that the execution of the
1178 FMD analysis had a flaw\footnote{A boolean variable was wrong in a
1179 configuration macro for FMD.} which caused the results to be lower than what they
1180 should be. The top panel of Figure \ref{prelimcomparison} shows a
1181 comparison between the distribution in Figure \ref{combineddndeta} and
1182 the preliminary (ALI-PREL-2536) shown at QM 2011. The top panel shows the same
1183 comparison with the proper FMD distribution instead of the incorrect
1184 one. It is clear that the agreement observed between VZERO, SPD,
1185 and FMD at QM 2011 does not hold with the FMD analysis run properly
1186 for nominal vertices.
1189 \begin{minipage}{0.5\linewidth}
1191 \includegraphics[keepaspectratio,width=\textwidth]{prelim_wrong150612}
1193 \begin{minipage}{0.5\linewidth}
1195 \includegraphics[keepaspectratio,width=\textwidth]{prelim_right150612}
1197 \caption{Left: Comparison of new combined $\dndeta$ to the data
1198 shown at QM 2011. Right: The same comparison with the properly run
1199 FMD analysis at nominal vertices (`FMD Hits'). The difference is
1200 clearly seen around $|\eta| \sim 2$.}
1201 \label{prelimcomparison}
1204 \subsection{Summary of Systematic Errors}
1205 Table \ref{combinedsyst} shows the various sources of systematic
1206 errors for the combined measurement of VZERO, SPD, and FMD collected
1207 from Table \ref{fmdsysterror}, \cite{maxime}, and
1208 \cite{ruben,Aamodt:2010cz}. The `common' section of the table refers to
1209 source of systematic errors identified as common in the different
1210 measurements. These errors were evaluated for the displaced vertices
1211 analysis in the following way:
1213 \item Centrality errors come from variation in the parameters used in
1214 the scaling of the ZEM signal (see \cite{maxime}).
1215 \item Material budget errors were estimated by analyzing a simulation
1216 and adding a weight of $0.9$ or $1.1$ to all physical processes except decays for all
1217 secondary particles. This approach was used in the absence of
1218 suitable ALICE simulation productions.
1219 \item $p_T$ weights were developed to assess the effect of the
1220 difference in $p_T$ spectra measured by ALICE and in the HIJING
1225 \begin{tabular}{|c|c|}
1227 Source of Error & Magnitude \\
1231 Centrality & 1-4\% \\
1233 $p_T$ weights (FMD+VZERO) & 2\% \\
1235 %Strangeness Enhancement & 1\% \\
1237 Material budget(FMD+VZERO) & 4\% \\
1243 Background Subtraction & 0.1\%-2\% \\
1245 Particle Mix & 1\% \\
1247 Weak Decays & 1 \% \\
1249 Extrapolation to zero $p$ & 2\% \\
1253 Fluctuation between rings & 3\% \\
1255 Normalization & 3\%-4\% \\
1259 Variation of Cuts & 2\% \\
1261 Calculation of Multiplicity & 3\% \\
1264 \caption[Combined Systematic Errors]{The table summarizes the
1265 systematic errors in the SPD\cite{ruben,Aamodt:2010cz}, VZERO\cite{maxime}, and FMD\cite{hhd:2009}.} \label{combinedsyst}
1267 The errors are obtained using variation of the quantities studied in
1268 MC simulations. In particular the studies of the dependence on the
1269 material budget are carried out with special MC simulations where the
1270 material density of ALICE is increased.
1271 \subsection{Technical Details}
1272 Here, the technical aspects of the analysis are described. The SPD
1273 analysis was done on run 137366, reconstruction pass 2 while the FMD
1274 and VZERO analysis were carried
1275 out on a total of 126 runs (46 with negative field and 80 with
1276 positive field) to obtain the necessary statistics for the displaced
1277 vertices. These runs were selected to be of good quality for VZERO, SPD, FMD, and
1278 ZDC. These data were also from pass 2 reconstruction.
1280 The AliRoot version for SPD is: \textbf{v5-03-24-AN}, for VZERO: \textbf{v5-03-28-AN}, and
1281 for FMD: \textbf{v5-03-26-AN}.
1283 For the analysis of the displaced vertices presented here the production LHC12c2 was used (the simulation was done with an anchor run for each field polarity). This production includes the latest version (as of July 2012) of the ALICE geometry and alignment.
1285 There is a twiki page for the paper using this analysis:
1286 \url{https://twiki.cern.ch/twiki/bin/viewauth/ALICE/PWGLFGeoPbPbdNdeta}.
1288 %% \currentpdfbookmark{Appendices}{Appendices}
1290 \section{Nomenclature}
1295 \begin{tabular}[t]{|lp{.8\textwidth}|}
1297 \textbf{Symbol}&\textbf{Description}\\
1299 \INEL & In--elastic event\\
1300 \INELONE & In--elastic event with at least one tracklet in the
1301 \SPD{} in the region $-1\le\eta\le1$\\
1302 \NSD{} & Non--single--diffractive event. Single diffractive
1303 events are events where one of the incident collision systems
1304 (proton or nucleus) is excited and radiates particles, but there
1305 is no other processes taking place\\
1307 $\NT{}$ & Number of events with a valid trigger\\
1308 $\NV{}$ & Number of events with a valid trigger \emph{and} a valid
1310 $\NA{}$ & Number of events with a valid trigger
1311 \emph{and} a valid vertex \emph{within} the selected vertex range.\\
1312 $\N{a,c,ac,e}{}$ & Number of events with background triggers $A$,
1313 $B$, $AC$, or $E$, \emph{and} a valid off-line trigger of the
1314 considered type. Background triggers are typically flagged with
1315 the trigger words \texttt{CINT1-A}, \texttt{CINT1-C},
1316 \texttt{CINT1-AC}, \texttt{CINT1-E}, or similar.\\
1318 $\mult{}$ & Charged particle multiplicity\\
1319 $\mult[,\text{primary}]$ & Primary charged particle multiplicity
1320 as given by simulations\\
1321 $\mult[,\text{\FMD{}}]$ & Number of charged particles that hit the
1322 \FMD{} as given by simulations\\
1323 $\mult[,t]$ & Number of charged particles in an \FMD{} strip as
1324 given by evaluating the energy response functions $F$\\
1326 $F$ & Energy response function (see \eqref{eq:energy_response})\\
1327 $\Delta_{mp}$ & Most probably energy loss\\
1328 $\xi$ & `Width' parameter of a Landau distribution\\
1329 $\sigma$ & Variance of a Gaussian distribution\\
1330 $a_i$ & Relative weight of the $i$--fold MIP peak in the energy
1333 $\Corners{}$ & Azimuthal acceptance of strip $t$\\
1334 $\SecMap{}$ & Secondary particle correction factor in $\etaphi$
1335 for a given vertex bin $v$\\
1336 $\DeadCh{}$ & Acceptance in $\etaphi$ for a given vertex bin $v$\\
1338 $\dndetadphi[incl,r,v,i]$ & Inclusive (primary \emph{and}
1339 secondary) charge particle density in event $i$ with vertex $v$,
1340 for \FMD{} ring $r$.\\
1341 $\dndetadphi[r,v,i]$ & Primary charged particle
1342 density in event $i$ with vertex $v$ for \FMD{} ring $r$. \\
1343 $\dndetadphi[v,i]$ & Primary charged particle density in event $i$
1345 $I_{v,i}(\eta)$ & $\eta$ acceptance of event $i$ with vertex $v$\\
1346 $I(\eta)$ & Integrated $\eta$ acceptance over $\NA$ events.
1347 Note, that this has a value of $\NA$ for $(\eta)$ bins where we
1348 have full coverage\\
1350 $X_t$ & Value $X$ for strip number $t$ (0-511 for inner rings,
1351 0-255 for outer rings)\\
1352 $X_r$ & Value $X$ for ring $r$ (where rings are \FMD{1i},
1353 \FMD{2i}, \FMD{2o}, \FMD{3o}, and \FMD{3i} in decreasing $\eta$
1355 $X_v$ & Value $X$ for vertex bin $v$ (typically 10 bins from -10cm
1357 $X_i$ & Value $X$ for event $i$\\
1360 \caption{Nomenclature used in this document}
1361 \label{tab:nomenclature}
1366 \section{Second pass example code}
1367 \label{app:exa_pass2}
1368 \lstset{basicstyle=\small\ttfamily,%
1369 keywordstyle=\color[rgb]{0.627,0.125,0.941}\bfseries,%
1370 identifierstyle=\color[rgb]{0.133,0.545,0.133}\itshape,%
1371 commentstyle=\color[rgb]{0.698,0.133,0.133},%
1372 stringstyle=\color[rgb]{0.737,0.561,0.561},
1373 emph={TH2D,TH1D,TFile,TTree,AliAODForwardMult},emphstyle=\color{blue},%
1374 emph={[2]dndeta,sum,norm},emphstyle={[2]\bfseries\underbar},%
1375 emph={[3]file,tree,mult,nV,nBg,nA,nT,i,gSystem},emphstyle={[3]},%
1378 \begin{lstlisting}[caption={Example 2\textsuperscript{nd} pass code to
1379 do $\dndeta$},label={lst:example},frame=single,captionpos=b]
1380 void Analyse(int mask=AliAODForwardMult::kInel,
1381 float vzLow=-10, float vzHigh=10, float trigEff=1)
1383 gSystem->Load("libANALYSIS.so"); // Load analysis libraries
1384 gSystem->Load("libANALYSISalice.so"); // General ALICE stuff
1385 gSystem->Load("libPWGLFforward2.so"); // Forward analysis code
1387 int nT = 0; // # of ev. w/trigger
1388 int nV = 0; // # of ev. w/trigger&vertex
1389 int nA = 0; // # of accepted ev.
1390 int nBg = 0; // # of background ev
1391 TH2D* sum = 0; // Summed hist
1392 AliAODForwardMult* mult = 0; // AOD object
1393 TFile* file = TFile::Open("AliAODs.root","READ");
1394 TTree* tree = static_cast<TTree*>(file->Get("aodTree"));
1395 tree->SetBranchAddress("Forward", &forward); // Set the address
1397 for (int i = 0; i < tree->GetEntries(); i++) {
1398 // Read the i'th event
1401 // Create sum histogram on first event - to match binning to input
1403 sum = static_cast<TH2D*>(mult->GetHistogram()->Clone("d2ndetadphi"));
1405 // Calculate beta=A+C-E
1406 if (mult->IsTriggerBits(mask|AliAODForwardMult::kA)) nBg++;
1407 if (mult->IsTriggerBits(mask|AliAODForwardMult::kC)) nBg++;
1408 if (mult->IsTriggerBits(mask|AliAODForwardMult::kE)) nBg--;
1410 // Other trigger/event requirements could be defined
1411 if (!mult->IsTriggerBits(mask)) continue;
1414 // Check if we have vertex and select vertex range (in centimeters)
1415 if (!mult->HasIpZ()) continue;
1418 if (!mult->InRange(vzLow, vzHigh) continue;
1421 // Add contribution from this event
1422 sum->Add(&(mult->GetHistogram()));
1425 // Get acceptance normalisation from underflow bins
1426 TH1D* norm = sum->ProjectionX("norm", 0, 0, "");
1427 // Project onto eta axis - _ignoring_underflow_bins_!
1428 TH1D* dndeta = sum->ProjectionX("dndeta", 1, -1, "e");
1429 // Normalize to the acceptance, and scale by the vertex efficiency
1430 dndeta->Divide(norm);
1431 dndeta->Scale(trigEff * nT/nV / (1 - nBg/nT), "width");
1432 // And draw the result
1437 \section{$\Delta E$ fits}
1438 \label{app:eloss_fits}
1440 \begin{figure}[htbp]
1442 \includegraphics[keepaspectratio,width=\textwidth]{eloss_fits}
1443 \caption{Summary of energy loss fits in each $\eta$ bin (see also
1444 \secref{sec:sub:sub:eloss_fits}).
1446 On the left side: Top panel shows the
1447 reduced $\chi^2$, second from the top shows the found
1448 scaling constant, 3\textsuperscript{rd} from the top is
1449 the most probable energy loss $\Delta_{mp}$, 4\textsuperscript{th}
1450 shows the width parameter $\xi$ of the Landau, and the
1451 5\textsuperscript{th} is the Gaussian width $\sigma$.
1452 $\Delta_{mp}$, $\xi$, and $\sigma$ have units of $\Delta E/\Delta
1455 On the right: The top panel shows the maximum number of
1456 multi--particle signals that where fitted, and the 4 bottom panels
1457 shows the weights $a_2,a_3,a_4,$ and $a_5$ for 2, 3, 4, and 5
1458 particle responses.}
1459 \label{fig:eloss_fits}
1463 \currentpdfbookmark{References}{References}
1464 \begin{thebibliography}{99}
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1479 %% \bibitem{Hancock:1983ry}
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1485 %% \bibitem{Hancock:1983fp}
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1504 %%CITATION = ARXIV:1012.1657;%%
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1511 %%CITATION = NUCL-EX/0112001;%%
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1517 %%CITATION = ARXIV:1011.1940;%%
1518 \end{thebibliography}
1522 % ispell-local-dictionary: "british"
1526 % LocalWords: tracklet diffractive IsTriggerBits AliAODForwardMult ProjectionX