1 \documentclass[11pt]{article}
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4 {\LARGE EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH}\\%
5 {\Large European Organization for Particle Physics}\\[2ex]%
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12 \includegraphics[keepaspectratio,width=.12\textwidth]{alicelogo}%
17 {\LARGE\bf Analysing the FMD data for $\dndeta$}%
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24 \hfill\includegraphics[keepaspectratio,width=.12\textwidth]{%
26 \hfill ALICE--INT--2012--040 v2\\
31 \author{Christian Holm
32 Christensen\thanks{\texttt{$\langle$cholm@nbi.dk$\rangle$}}\quad\&\quad
33 Hans Hjersing Dalsgaard\thanks{\texttt{$\langle$canute@nbi.dk$\rangle$}}\\
34 Niels Bohr Institute\\
35 University of Copenhagen}
38 \pdfbookmark{Analysing the FMD data for dN/deta}{top}
42 \section{Introduction}
44 This document describes the steps performed in the analysis of the
45 charged particle multiplicity in the forward pseudo--rapidity regions
46 with the \FMD{} detector \cite{FWD:2004mz,cholm:2009}. The document
47 also include a summary (see section \ref{prelim}) of the request for
48 preliminary figures for the measurement of $\dndeta$ with
49 SPD\cite{ruben,Aamodt:2010cz}, VZERO\cite{maxime}, and FMD.
50 % The primary detector used for this is the \FMD{}
52 The \FMD{} is organised in 3 \emph{sub--detectors} \FMD{1}, \FMD{2},
53 and \FMD{3}, each consisting of 1 (\FMD{1}) or 2 (\FMD{2} and~3)
54 \emph{rings}. The rings fall into two types: \emph{Inner} or
55 \emph{outer} rings. Each ring is in turn azimuthally divided into
56 \emph{sectors}, and each sector is radially divided into
57 \emph{strips}. How many sectors, strips, as well as the $\eta$
58 coverage is given in \tablename~\ref{tab:fmd:overview}.
62 \caption{Physical dimensions of Si segments and strips.}
63 \label{tab:fmd:overview}
65 \begin{tabular}{|c|cc|cr@{\space--\space}l|r@{\space--\space}l|}
67 \textbf{Sub--detector/} &
71 \multicolumn{2}{c|}{\textbf{$r$}} &
72 \multicolumn{2}{c|}{\textbf{$\eta$}} \\
77 \multicolumn{2}{c|}{\textbf{range [cm]}} &
78 \multicolumn{2}{c|}{\textbf{coverage}} \\
80 FMD1i & 20& 512& 320 & 4.2& 17.2& 3.68& 5.03\\
81 FMD2i & 20& 512& 83.4& 4.2& 17.2& 2.28& 3.68\\
82 FMD2o & 40& 256& 75.2& 15.4& 28.4& 1.70& 2.29\\
83 FMD3i & 20& 512& -75.2& 4.2& 17.2&-2.29& -1.70\\
84 FMD3o & 40& 256& -83.4& 15.4& 28.4&-3.40& -2.01\\
90 The \FMD{} \ESD{} object contains the scaled energy deposited $\Delta
91 E/\Delta E_{mip}$ for each of the 51,200 strips. This is determined
92 in the reconstruction pass. The scaling to $\Delta E_{mip}$ is done
93 using calibration factors extracted in designated pulser runs. In
94 these runs, the front-end electronics is pulsed with an increasing
95 known pulse size, and the conversion factor from ADC counts to $\Delta
96 E_{mip}$ is determined \cite{cholm:2009}.
98 The \SPD{} is used for determination of the position of the primary
99 interaction point except in the case of displaced vertex analysis as
100 discussed in section \ref{sec:sub:sub:dispvtx}.
102 The analysis is performed as a two--step process.
104 \item The Event--Summary--Data (\ESD{}) is processed event--by--event
105 and passed through a number of algorithms, and
106 $\dndetadphi$ for each event is output to an Analysis--Object--Data
107 (\AOD{}) tree (see \secref{sec:gen_aod}).
108 \item The \AOD{} data is read in and the sub--sample of the data under
109 investigation is selected (e.g., \INEL{}, \INELONE{}, \NSD{} in p+p data, or
110 some centrality class in Pb+Pb data) and the $\dndetadphi$ histogram read for
111 those events to build up $\dndeta$ (see \secref{sec:ana_aod}).
113 The details of each step above will be expanded upon in the
116 In Appendix~\ref{app:nomen} is an overview of the nomenclature used in
119 \section{Generating $\dndetadphi[i]$ event--by--event}
122 When reading in the \ESD{}s and generating the $\dndetadphi$
123 event--by--event the following steps are taken (in order) for each
124 event $i$ and FMD ring $r$.
126 \item[Event inspection] The global properties of the event is
127 determined, including the trigger type and primary interaction
128 point\footnote{`Vertex' and `primary interaction point' will be used
129 interchangeably in the text, since there is no ambiguity with
130 particle production vertex in this analysis.} $z$ coordinate (see
131 \secref{sec:sub:event_inspection}).
132 \item[Sharing filter] The \ESD{} object is read in and corrected for
133 sharing. The result is a new \ESD{} object (see
134 \secref{sec:sub:sharing_filter}).
135 \item[Density calculator] The (possibly un--corrected) \ESD{} object
136 is then inspected and an inclusive (primary \emph{and} secondary
137 particles), per--ring charged particle density
138 $\dndetadphi[incl,r,v,i]$ is made. This calculation depends in
139 general upon the interaction vertex position along the $z$ axis
140 $v_z$ (see \secref{sec:sub:density_calculator}).
141 \item[Corrections] The 5 (one for each FMD ring)
142 $\dndetadphi[incl,r,v,i]$ are corrected for secondary production and
143 acceptance. The correction for the secondary particle production is
144 highly dependent on the vertex $z$ coordinate. The result is a
145 per--ring, charged primary particle density $\dndetadphi[r,v,i]$
146 (see \secref{sec:sub:corrector}).
147 \item[Histogram collector] Finally, the 5 $\dndetadphi[r,v,i]$ are
148 summed into a single $\dndetadphi[v,i]$ histogram, taking care of
149 the overlaps between the detector rings. In principle, this
150 histogram is independent of the vertex, except that the
151 pseudo--rapidity range, and possible holes in that range, depends on
152 $v_z$ --- or rather the bin in which the $v_z$ falls (see
153 \secref{sec:sub:hist_collector}).
156 Each of these steps will be detailed in the following.
158 \subsection{Event inspection}
159 \label{sec:sub:event_inspection}
161 The first thing to do, is to inspect the event for triggers. A number
162 of trigger bits, like \INEL{} (Minimum Bias for Pb+Pb), \INELONE{},
163 \NSD{}, and so on is then propagated to the \AOD{} output.
165 Just after the sharing filter (described below) but before any further
166 processing, the vertex information is queried. If there is no vertex
167 information, or if the vertex $z$ coordinate is outside the
168 pre--defined range, then no further processing of that event takes
171 \subsubsection{Displaced Vertices}
172 \label{sec:sub:sub:dispvtx}
174 The analysis can be set up to run on the `displaced vertices' that
175 occur during LHC Pb+Pb running. Details on the displaced vertices, and
176 their selection can be found in the VZERO analysis note \cite{maxime}.
178 \subsection{Sharing filter}
179 \label{sec:sub:sharing_filter}
181 A particle originating from the vertex can, because of its incident
182 angle on the \FMD{} sensors traverse more than one strip (see
183 \figref{fig:share_fraction}). This means that the energy loss of the
184 particle is distributed over 1 or more strips. The signal in each
185 strip should therefore possibly be merged with its neighboring strip
186 signals to properly reconstruct the energy loss of a single particle.
190 \includegraphics[keepaspectratio,height=3cm]{share_fraction}
191 \caption{A particle traversing 2 strips and depositing energy in
193 \label{fig:share_fraction}
196 The effect is most pronounced in low--flux\footnote{Events with a low
197 hit density.} events, like proton--proton collisions or peripheral
198 Pb--Pb collisions, while in high--flux events the hit density is so
199 high that most likely each and every strip will be hit and the effect
200 cancels out on average.
202 Since the particles travel more or less in straight lines toward the
203 \FMD{} sensors, the sharing effect is predominantly in the $r$ or
204 \emph{strip} direction. Only neighbouring strips in a given sector are
205 therefore investigated for this effect.
207 Algorithm~\ref{algo:sharing} is applied to the signals in a given
210 \begin{algorithm}[htpb]
211 \belowpdfbookmark{Algorithm 1}{algo:sharing}
212 \SetKwData{usedThis}{current strip used}
213 \SetKwData{usedPrev}{previous strip used}
214 \SetKwData{Output}{output}
215 \SetKwData{Input}{input}
216 \SetKwData{Nstr}{\# strips}
217 \SetKwData{Signal}{current}
218 \SetKwData{Eta}{$\eta$}
219 \SetKwData{prevE}{previous strip signal}
220 \SetKwData{nextE}{next strip signal}
221 \SetKwData{lowFlux}{low flux flag}
222 \SetKwFunction{SignalInStrip}{SignalInStrip}
223 \SetKwFunction{MultiplicityOfStrip}{MultiplicityOfStrip}
224 \usedThis $\leftarrow$ false\;
225 \usedPrev $\leftarrow$ false\;
226 \For{$t\leftarrow1$ \KwTo \Nstr}{
227 \Output${}_t\leftarrow 0$\;
228 \Signal $\leftarrow$ \SignalInStrip($t$)\;
230 \uIf{\Signal is not valid}{
231 \Output${}_t \leftarrow$ invalid\;
233 \uElseIf{\Signal is 0}{
234 \Output${}_t \leftarrow$ 0\;
237 \Eta$\leftarrow$ $\eta$ of \Input${}_t$\;
238 \prevE$\leftarrow$ 0\;
239 \nextE$\leftarrow$ 0\;
241 \prevE$\leftarrow$ \SignalInStrip($t-1$)\;
244 \nextE$\leftarrow$ \SignalInStrip($t+1$)\;
246 \Output${}_t\leftarrow$
247 \MultiplicityOfStrip(\Signal,\Eta,\prevE,\nextE,\\
248 \hfill\lowFlux,$t$,\usedPrev,\usedThis)\;
251 \caption{Sharing correction}
255 Here the function \FuncSty{SignalInStrip}($t$) returns the properly
256 path--length corrected signal in strip $t$. The function
257 \FuncSty{MultiplicityOfStrip} is where the real processing takes
258 place (see page \pageref{func:MultiplicityOfStrip}).
260 \begin{function}[htbp]
261 \belowpdfbookmark{MultiplicityOfStrip}{func:MultiplicityOfStrip}
262 \caption{MultiplicityOfStrip(\DataSty{current},$\eta$,\DataSty{previous},\DataSty{next},\DataSty{low
263 flux flag},\DataSty{previous signal used},\DataSty{this signal
265 \label{func:MultiplicityOfStrip}
266 \SetKwData{Current}{current}
267 \SetKwData{Next}{next}
268 \SetKwData{Previous}{previous}
269 \SetKwData{lowFlux}{low flux flag}
270 \SetKwData{usedPrev}{previous signal used}
271 \SetKwData{usedThis}{this signal used}
272 \SetKwData{lowCut}{low cut}
273 \SetKwData{total}{Total}
274 \SetKwData{highCut}{high cut}
275 \SetKwData{Eta}{$\eta$}
276 \SetKwFunction{GetHighCut}{GetHighCut}
277 \If{\Current is very large or \Current $<$ \lowCut} {
278 \usedThis $\leftarrow$ false\;
279 \usedPrev $\leftarrow$ false\;
283 \usedThis $\leftarrow$ false\;
284 \usedPrev $\leftarrow$ true\;
287 \highCut $\leftarrow$ \GetHighCut($t$,\Eta)\;
288 %\If{\Current $<$ \Next and \Next $>$ \highCut and \lowFlux set}{
289 % \usedThis $\leftarrow$ false\;
290 % \usedPrev $\leftarrow$ false\;
293 \total $\leftarrow$ \Current\;
294 \lIf{\lowCut $<$ \Previous $<$ \highCut and not \usedPrev}{
295 \total $\leftarrow$ \total + \Previous\;
297 \If{\lowCut $<$ \Next $<$ \highCut}{
298 \total $\leftarrow$ \total + \Next\;
299 \usedThis $\leftarrow$ true\;
302 \usedPrev $\leftarrow$ true\;
305 \usedPrev $\leftarrow$ false\;
306 \usedThis $\leftarrow$ false\;
310 Here, the function \FuncSty{GetHighCut} (see below) evaluates a fit to the energy
311 distribution in the specified $\eta$ bin (see also
312 \secref{sec:sub:density_calculator}). It returns
316 where $\Delta_{mp}$ is the most probable energy loss, and $w$ is the
317 width of the Landau distribution.
319 The \KwSty{if} in line 5, says that if the previous strip was merged
320 with current one, and the signal of the current strip was added to
321 that, then the current signal is set to 0, and we mark it as used for
322 the next iteration (\DataSty{previous signal used}$\leftarrow$true).
324 % The \KwSty{if} in line 10 checks if the current signal is smaller than
325 % the next signal, if the next signal is larger than the upper cut
326 % defined above, and if we have a low--flux event\footnote{Note, that in
327 % the current implementation there are never any low--flux events.}.
328 % If that condition is met, then the current signal is the smaller of
329 % two possible candidates for merging, and it should be merged into the
330 % next signal. Note, that this \emph{only} applies in low--flux events.
333 we test if the previous signal lies between our low and
334 high cuts, and if it has not been marked as being used. If so, we add
335 it to our current signal.
337 The next \KwSty{if} on line 12 % 16
338 checks if the next signal is within our
339 cut bounds. If so, we add that signal to the current signal and mark
340 it as used for the next iteration (\DataSty{this signal
341 used}$\leftarrow$true). It will then be put to zero on the next
342 iteration by the condition on line 6.
344 Finally, if our signal is still larger than 0, we return the signal
345 and mark this signal as used (\DataSty{previous signal
346 used}$\leftarrow$true) so that it will not be used in the next
347 iteration. Otherwise, we mark the current signal and the next signal
348 as unused and return a 0.
351 \subsection{Density calculator}
352 \label{sec:sub:density_calculator}
354 The density calculator loops over all the strip signals in the sharing
355 corrected\footnote{The sharing correction can be disabled, in which
356 case the density calculator uses the input \ESD{} signals.} \ESD{}
357 and calculates the inclusive (primary + secondary) charged particle
358 density in pre--defined $\etaphi$ bins.
360 \subsubsection{Inclusive number of charged particles: Energy Fits}
361 \label{sec:sub:sub:eloss_fits}
363 The number charged particles in a strip $\mult[,t]$ is calculated
364 using multiple Landau-like distributions fitted to the energy loss
365 spectrum of all strips in a given $\eta$ bin.
367 \Delta_{i,mp} &= i (\Delta_{1,mp}+ \xi_1 \log(i))\nonumber\\
368 \xi_i &= i\xi_1\nonumber\\
369 \sigma_i &= \sqrt{i}\sigma_1\nonumber\\
370 \mult[,t] &= \frac{\sum_i^{N_{max}}
371 i\,a_i\,F(\Delta_t;\Delta_{i,mp},\xi_i,\sigma_i)}{
372 \sum_i^{N_{max}}\,a_i\,F(\Delta_t;\Delta_{i,mp},\xi_i,\sigma_i)}\quad,
374 where $F(x;\Delta_{mp},\xi,\sigma)$ is the evaluation of the Landau
375 distribution $f_L$ with most probable value $\Delta_{mp}$ and width
376 $\xi$, folded with a Gaussian distribution with spread $\sigma$ at the
377 energy loss $x$ \cite{nim:b1:16,phyrev:a28:615}.
379 \label{eq:energy_response}
380 F(x;\Delta_{mp},\xi,\sigma) = \frac{1}{\sigma \sqrt{2 \pi}}
381 \int_{-\infty}^{+\infty} d\Delta' f_{L}(x;\Delta',\xi)
382 \exp{-\frac{(\Delta_{mp}-\Delta')^2}{2\sigma^2}}\quad,
384 where $\Delta_{1,mp}$, $\xi_1$, and $\sigma_1$ are the parameters for
385 the first MIP peak, $a_1=1$, and $a_i$ is the relative weight of the
386 $i$-fold MIP peak. The parameters $\Delta_{1,mp}, \xi_1,
387 \sigma_1, \mathbf{a} = \left(a_2, \ldots a_{N_{max}}\right)$ are
390 F_j(x;C,\Delta_{mp},\xi,\sigma,\mathbf{a}) = C
391 \sum_{i=1}^{j} a_i F(x;\Delta_{i,mp},\xi_{i},\sigma_i)
393 for increasing $j$ to the energy loss spectra in separate $\eta$ bins.
394 The fit procedure is stopped when for $j+1$: (the default values for
395 each value are included below)
397 \item the reduced $\chi^2$ exceeds a certain threshold (usually 20), or
398 \item the relative error $\delta p/p$ of any parameter of the fit
399 exceeds a certain threshold (usually 0.12), or
400 \item when the weight $a_j+1$ is smaller than some number (typically
403 $N_{max}$ is then set to $j$. Examples of the result of these fits
404 are given in \figref{fig:eloss_fits} in Appendix~\ref{app:eloss_fits}.
405 \subsubsection{Inclusive number of charged particles: Poisson Approach}
406 \label{sec:sub:sub:poisson}
407 Another approach to the calculation of the number of charged particles
408 is using Poisson statistics. This is the default choice because it is
409 less sensitive to the stability of the fits required for the energy
411 Assume that in a region of the FMD % where
413 %is azimuthally uniform in $\eta$ intervals it
415 distributed according to a Poisson distribution. This means that the
416 probability of $\mult=n$ becomes:
418 P(n) = \frac{\mu^n e^{-\mu}}{n!} \label{eq:PoissonDef}
420 In particular the measured occupancy, $\mu_{meas}$, is the probability
421 of any number of hits, thus using \eqref{eq:PoissonDef} :
423 \mu_{meas} = 1 - P(0) = 1 - e^{-\mu }
424 %\Rightarrow \mu = \ln
425 %(1 - \mu_{meas})^{-1} \label{eq:PoissonDef2}
430 (1 - \mu_{meas})^{-1} \label{eq:PoissonDef2}
432 The mean number of particles in a hit strip becomes:
434 C &=& \frac{\sum_{n>0} n P(n>0)}{\sum_{n>0} P(n>0)} \nonumber \\
435 &=& \frac{e^{-\mu}}{1-e^{-\mu}} \mu \sum \frac{\mu^n}{n!}
437 &=& \frac{e^{-\mu}}{1-e^{-\mu}} \mu e^{\mu} \nonumber \\
438 &=& \frac{\mu}{1-e^{-\mu}}
440 %While $\mu$ can be calculated analytically for practical purposes we
441 With $\mu$ defined in \eqref{eq:PoissonDef2} this calculation is
442 carried out per event in
443 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
444 %$\mu_{meas}^{region}$ to be the measured occupancy
446 $\mult$ for a hit strip ($N_{hits} \equiv 1$) in that region becomes:
448 \mult = N_{hits} \times C = 1 \times C = C
450 Where C is calculated using $\mu_{meas}^{region}$.
452 The Poisson method and the energy fits method have been compared in
453 \cite{hhd:2009} where it is found that the two methods are in good
454 agreement. The residual difference between the methods contributes to
455 the systematic error.
457 \subsubsection{Azimuthal Acceptance}
459 Before the signal $\mult[,t]$ can be added to the $\etaphi$
460 bin in one of the 5 per--ring histograms, it needs to be corrected for
461 the $\varphi$ acceptance of the strip.
463 The sensors of the \FMD{} are not perfect arc--segments --- the two
464 top corners are cut off to allow the largest possible sensor on a 6''
465 Si-wafer. This means, however, that the strips in these outer
466 regions do not fully cover $2\pi$ in azimuth, and there is therefore a
467 need to correct for this limited acceptance.
469 The acceptance correction is only applicable where the strip length
470 does not cover the full sector. This is the case for the outer strips
471 in both the inner and outer type rings. The acceptance correction is
475 \Corners{} &= \frac{l_t}{\Delta\varphi}\quad
477 where $l_t$ is the strip length in radians at constant $r$, and
478 $\Delta\varphi$ is $2\pi$ divided by the number of sectors in the
479 ring (20 for inner type rings, and 40 for outer type rings).
481 Note, that this correction is a hardware--related correction, and does
482 not depend on the properties of the collision (e.g., primary vertex
485 The final $\etaphi$ content of the 5 output vertex dependent,
486 per--ring histograms of the inclusive charged particle density is then
490 \dndetadphi[incl,r,v,i\etaphi] &= \sum_t^{t\in\etaphi}
491 \mult[,t]\,\Corners{}
493 where $t$ runs over the strips in the $\etaphi$ bin.
495 \subsection{Corrections}
496 \label{sec:sub:corrector}
498 The corrections code receives the five vertex dependent,
499 per--ring histograms of the inclusive charged particle density
500 $\dndetadphi[incl,r,v,i]$ from the density calculator and applies
503 \subsubsection{Secondary correction}
505 %% hHits_FMD<d><r>_vtx<v>
506 %% hCorrection = -----------------------
507 %% hPrimary_FMD_<r>_vtx<v>
510 %% - hPrimary_FMD_<r>_vtx<vtx> is 2D of eta,phi for all primary ch
512 %% - hHits_FMD<d><r>_vtx<v> is 2D of eta,phi for all track-refs that
513 %% hit the FMD - The 2D version of hMCHits_nocuts_FMD<d><r>_vtx<v>
515 This is a 2 dimensional histogram generated from simulations, as the
516 ratio of primary particles to the total number of particles that fall
517 within an $\etaphi$ bin for a given vertex bin
522 \frac{\sum_i^{\NV[,v]}\mult[,\text{primary},i]\etaphi}{
523 \sum_i^{\NV[,v]}\mult[,\text{\FMD{}},i]\etaphi}\quad,
525 where $\NV[,v]$ is the number of events with a valid trigger and a
526 vertex in bin $v$, and $\mult[,\FMD{},i]$ is the total number of
527 charged particles that hit the \FMD{} in event $i$ in the specified
528 $\etaphi$ bin and $\mult[,\text{primary},i]$ is number of
529 primary charged particles in event $i$ within the specified
532 $\mult[,\text{primary}]\etaphi$ is given by summing over the
533 charged particles labelled as primaries \emph{at the time of the
534 collision} as defined in the simulation code. That is, it is the
535 number of primaries within the $\etaphi$ bin at the collision
536 point --- not at the \FMD{}.
538 $\SecMap$ varies from $\approx 1.5$ for the most forward bins to
539 $\approx 3$ for the more central bins. Figure \ref{secondaries} shows
540 the $\dndeta$ of secondaries from various sources assessed with MC
541 simulations to give an idea of the magnitude of the effects of
545 \includegraphics[keepaspectratio,width=\textwidth]{%
547 \caption{$\dndeta$ for secondaries and primaries in the FMD. The
548 same plot for the SPD inner layer is included for comparison.}
552 %For pp, different event
553 %generators were used and found to give compatible results within
555 For pp, at least some millions of events must be
556 accumulated to reach satisfactory statistics. For Pb--Pb where the
557 general hit density is larger, reasonable statistics can be achieved
558 with less simulated data.
560 \subsubsection{Acceptance due to dead channels}
562 Some of the strips in the \FMD{} have been marked up as \emph{dead},
563 meaning that they are not used in the reconstruction or analysis.
564 This leaves holes in the acceptance of each defined $\etaphi$
565 which need to be corrected for.
567 Dead channels are marked specially in the \ESD{}s with the flag
568 \textit{Invalid Multiplicity}. This is used in the analysis to build
569 up and event--by--event acceptance correction in each $\etaphi$
570 bin by calculating the ratio
572 \label{eq:dead_channels}
574 \frac{\sum_t^{t\in\etaphi}\left\{\begin{array}{cl}
575 1 & \text{if not dead}\\
577 \end{array}\right.}{\sum_t^{t\in\etaphi} 1}\quad,
579 where $t$ runs over the strips in the $\etaphi$ bin. This correction
580 is obviously $v_z$ dependent since the $\etaphi$ bin to which a strip $t$
581 corresponds to depends on its position relative to the primary vertex.
583 Alternatively, pre--made acceptance factors can be used. These are
584 made from the off-line conditions database (\OCDB{}).
586 The 5 output vertex dependent, per--ring histograms of the primary
587 charged particle density is then given by
589 \dndetadphi[r,v,i\etaphi] &=
590 \SecMap{} \frac{1}{\DeadCh{}}\dndetadphi[incl,r,v,i\etaphi]
593 \subsection{Histogram collector}
594 \label{sec:sub:hist_collector}
596 The histogram collector collects the information from the 5 vertex
597 dependent, per--ring histograms of the primary charged particle
598 density $\dndetadphi[r,v,i]$ into a single vertex dependent histogram
599 of the charged particle density $\dndetadphi[v,i]$.
601 To do this, it first calculates, for each vertex bin, the $\eta$ bin
602 range to use for each ring. It investigates the secondary correction
603 maps $\SecMap{}$ to find the edges of each map. The edges are given
604 by the $\eta$ range where $\SecMap{}$ is larger than some
605 threshold\footnote{Typically $t_s\approx 0.1$.} $t_s$. The code
606 applies safety margin of a number of bins, $N_{cut}$\footnote{Typically
607 $N_{cut}=1$.}, to ensure that the data selected does not have too
608 large corrections associated with it.
610 It then loops over the bins in the defined $\eta$ range and sums the
611 contributions from each of the 5 histograms. In the $\eta$ ranges
612 where two rings overlap, the collector calculates the average and adds
613 the errors in quadrature\footnote{While not explicitly checked, it was
614 found that the histograms agrees within error bars in the
617 The output vertex dependent histogram of the primary
618 charged particle density is then given by
621 \dndetadphi[v,i\etaphi] &=
622 \frac{1}{N_{r\in\etaphi}}\sum_{r}^{r\in\etaphi}
623 \dndetadphi[r,v,i\etaphi]\\
624 \delta\left[\dndetadphi[v,i\etaphi]\right] &=
625 \frac{1}{N_{r\in\etaphi}}\sqrt{\sum_{r}^{r\in\etaphi}
626 \delta\left[\dndetadphi[r,v,i\etaphi]\right]^2}
629 where $N_{r\in\etaphi}$ is the number of overlapping histograms
630 in the given $\etaphi$ bin.
632 The histogram collector stores the found $\eta$ ranges in the
633 underflow bin of the histogram produced. The content of the overflow
638 \frac{1}{N_{r\in(\eta)}}
639 \sum_{r}^{r\in(\eta)} \left\{\begin{array}{cl}
640 0 & \eta \text{\ bin not selected}\\
641 1 & \eta \text{\ bin selected}
642 \end{array}\right.\quad,
644 where $N_{r\in(\eta)}$ is the number of overlapping histograms in the
645 given $\eta$ bin. The subscript $v$ indicates that the content
646 depends on the current vertex bin of event $i$.
648 \section{Building the final $\dndeta$}
651 To build the final $\dndeta$ distribution it is enough to sum
652 \eqref{eq:superhist} and \eqref{eq:overflow} over all accepted
653 events, $\NA$, and correct for the acceptance $I(\eta)$
655 \dndetadphi[\etaphi] &= \sum_i^{\NA}\dndetadphi[i,v\etaphi]\\
656 I(\eta) &= \sum_i^{\NA}I_{i,v}(\eta)\quad.
658 Note, that $I(\eta)\le\NA$.
660 We then need to normalise to the total number of events $N_X$, given
663 \N{X}{} &= \frac{1}{\epsilon_X}\left[\NA + \alpha(\NnotV -
664 \beta)\right] \label{eq:fulleventnorm}\\
665 & = \frac{1}{\epsilon_X}\left[\NA + \frac{\NA}{\NV}(\NT-\NV{} -
666 \beta)\right]\nonumber \\
667 & =\frac{1}{\epsilon_X}\NA\left[1+\frac{1}{\epsilon_V}-1-
668 \frac{\beta}{\NV}\right]\nonumber\\
669 & = \frac{1}{\epsilon_X}\frac{1}{\epsilon_V}\NA
670 \left(1-\frac{\beta}{\NT{}}\right)\nonumber
674 \item[$\epsilon_X$] is the trigger efficiency for type
675 $X\in[\text{\INEL},\text{\INELONE},\text{\NSD} for p+p data and MB
677 \item[$\epsilon_V=\frac{\NV{}}{\NT{}}$] is the vertex efficiency
678 evaluated over the data.
679 \item[$\NA$] is the number of events with a trigger \emph{and} a valid
680 vertex in the selected range
681 \item[$\NV{}$] is the number of events with a trigger \emph{and} a valid
683 \item[$\NT$] is the number of events with a trigger.
684 \item[$\NnotV{}=\NT-\NV{}$] is the number of events with a trigger
685 \emph{but no} valid vertex
686 \item[$\alpha=\frac{\NA}{\NV}$] is the fraction of accepted events of
687 the total number of events with a trigger and valid vertex.
688 \item[$\beta=\N{a}{}+\N{c}{}-\N{e}{}$] is the number of background
689 events \emph{with} a valid off-line trigger. This formula is the
690 simplest case of one bunch crossing per trigger/background
691 class. For more complicated collision setups the fractions in the
694 The two terms under the parenthesis in \eqref{eq:fulleventnorm} refers
695 to the observed number of event $\NA$, and the events missed because
696 of no vertex reconstruction. Note, for $\beta\ll\NT{}$
697 \eqref{eq:fulleventnorm} reduces to the simpler expression
699 \N{X}{} = \frac1{\epsilon_X}\frac1{\epsilon_V}\NA{}
701 The trigger efficiency $\epsilon_X$ for a given trigger type $X$ is
702 evaluated from simulations as
704 \epsilon_X = \frac{\N{X\wedge \text{T}}{}}{\N{X}{}}\quad,
706 that is, the ratio of number of events of type $X$ with a
707 corresponding trigger to the number of events of type $X$.
709 The final event--normalised charged particle density then becomes
711 \frac{1}{N}\frac{dN_{\text{ch}}}{d\eta} &=
712 \frac{1}{\N{X}{}} \int_0^{2\pi} d\varphi
713 \frac{\dndetadphi[\etaphi]}{I(\eta)}
714 \label{eq:eventnormdndeta}
717 If the trigger $X$ introduces a bias on the measured number of events,
718 then \eqref{eq:eventnormdndeta} need to be modified to
720 \frac{1}{N}\frac{dN_{\text{ch}}}{d\eta} &=
721 \frac{1}{\N{X}{}} \int_0^{2\pi} d\varphi
722 \frac{\frac{1}{B\etaphi}\dndetadphi[\etaphi]}{I(\eta)}
723 \label{eq:eventnormdndeta2}\quad,
725 where $B\etaphi$ is the bias correction. This is typically
726 calculated from simulations using the expression
728 B\etaphi = \frac{\frac{1}{\N{X\wedge
729 \text{T}}{}}\sum_i^{\N{X\wedge \text{T}}{}}
730 \mult[,\text{primary}]\etaphi}{\frac{1}{\N{X}{}}\sum_i^{\N{X}{}}
731 \mult[,\text{primary}]\etaphi}
734 \section{Systematic Errors} \label{fmdsysterror}
737 \begin{tabular}{|c|c|c|}
739 Effect & Magnitude in Pb+Pb analysis & Magnitude in p+p
742 Variation of the cuts in sec. \ref{sec:sub:sharing_filter} & 2\% & 3\% \\
744 Calculation of $\mult$ & 3\% & 4\% \\
746 Material budget & 7 \% & 7 \% \\
748 Generator & 2\% & 2\% \\
750 Vertex and trigger bias & N/A & 3\% \\
752 Centrality & 1\% --6\% & N/A \\
754 Normalization & N/A & 1.3\% - 3\% \\
757 Total in quadrature & 8.2\% -- 10.1\% & 9.4 \% -- 9.8\% \\
760 \caption[Systematic Errors in the FMD]{The table summarizes the
761 systematic errors in the FMD including the total systematic error
762 obtained by addition in quadrature.} \label{systerrors}
764 The systematic errors on the $\dndeta$ measurement are discussed in detail in
765 \cite{hhd:2009}. The results for the systematic errors in p+p and
766 Pb+Pb data are shown in table \ref{systerrors}. A short summary of the elements of the table is given here:
768 \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.
769 \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.
770 \item The systematic error on the material budget description was found from simulations with $\pm 10 \%$ increased density.
771 \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).
772 \item The systematic error on the centrality selection was obtained from variations of the different methods for measuring centrality.
775 \section{Using the per--event $\dndetadphi[i,v]$ histogram for other
778 \subsection{Multiplicity distribution}
780 To build the multiplicity distribution for a given $\eta$ range
781 $[\eta_1,\eta_2]$, one needs to find the total multiplicity in that
782 $\eta$ range for each event. To do so, one should sum the
783 $\dndetadphi[i,v]$ histogram over all $\varphi$ and in the selected
786 n'_{i[\eta_1,\eta_2]}, &= \int_{\eta_1}^{\eta_2}d\eta\int_0^{2\pi}d\varphi
787 \dndetadphi[i,v]\quad.\nonumber
789 However, $n'_i$ is not corrected for the coverage in $\eta$ for the
790 particular vertex range $v$. One therefor needs to correct for the
791 number of missing bins in the range $[\eta_1,\eta_2]$. Suppose
792 $[\eta_1,\eta_2]$ covers $N_{[\eta_1,\eta_2]}$ $\eta$ bins, then the acceptance
793 correction is given by
795 A_{i,[\eta_1,\eta_2]} = \frac{N_{[\eta_1,\eta_2]}}{\int_{\eta_1}^{\eta_2}d\eta\,
796 I_{i,v}(\eta)}\quad.\nonumber
798 The per--event multiplicity is then given by
800 n_{i,[\eta_1,\eta_2]} &= A_{i,[\eta_1,\eta_2]}\,n'_{i,[\eta_1,\eta_2]}\nonumber\\
801 &= \frac{N_{[\eta_1,\eta_2]}}{\int_{\eta_1}^{\eta_2}\eta
802 I_{i,v}(\eta)} \int_{\eta_1}^{\eta_2}d\eta\int_0^{2\pi}d\varphi
807 \subsection{Forward--Backward correlations}
809 To do forward--backward correlations, one need to calculate
810 $n_{i,[\eta_1,\eta_2]}$ as shown in \eqref{eq:event_n} in two bins
811 $n_{i,[\eta_1,\eta_2]}$ and $n_{i,[-\eta_2,-\eta_1]}$ \textit{e.g.},
812 $n_{i,f}=n_{i,[-3,-1]}$ and $n_{i,b}=n_{i,[1,3]}$.
815 \section{Some results}
817 %% \figurename{}s \ref{fig:1} to \ref{fig:3} shows some results.
818 Figures below show some examples \cite{hhd:2009}. Note these are not
822 \includegraphics[keepaspectratio,width=\textwidth]{results_ppdndeta}
823 \caption{$\dndeta$ for pp for \INEL{} events at
824 $\sqrt{s}=\GeV{900}$, $\sqrt{s}=\TeV{2.76}$, and $\sqrt{s}=\TeV{7}$
825 $\cm{-10}\le v_z\le\cm{10}$, rebinned by a factor 5 \cite{hhd:2009}.
827 % shows the ratio of ALICE data to UA5, and the bottom panel shows
828 % the ratio of the right (positive) side to the left (negative) side
829 % of the forward $\dndeta$.
835 \includegraphics[keepaspectratio,width=\textwidth]{results_PbPbdndeta}
836 \caption{$\dndeta$ for Pb+Pb for Minimum Bias events at
837 $\sqrt{s_{NN}}=\TeV{2.76}$ $\cm{-10}\le v_z\le\cm{10}$, rebinned by a
838 factor 5 in 10 centrality intervals \cite{hhd:2009}.
840 % shows the ratio of ALICE data to UA5, and the bottom panel shows
841 % the ratio of the right (positive) side to the left (negative) side
842 % of the forward $\dndeta$.
850 \section{Analysis for QM 2012 and Paper} \label{prelim}
851 \subsection{Analysis}
853 Following the development of the displaced vertex technique for VZERO
854 \cite{maxime} it was decided also to attempt such an analysis with the
855 FMD using exactly the same event selection and centrality selection as
858 The analysis described in this note was used successfully on these
859 special events. Three detectors contribute to this measurement: SPD
860 with tracklets covering $-2<\eta<2$ \cite{ruben,Aamodt:2010cz}, VZERO
861 covering $-3<\eta<-1.25$ and $1.25<\eta<5.25$, and FMD covering
862 $-5<\eta<-1.25$ and $1.25<\eta<5.5$. The extended coverage of the
863 VZERO and FMD comes from the positions of the displaced vertices. The
864 full pseudorapidity coverage of the combined measurement is
867 To combine the measurements the individual measurements were weighted
868 by their systematic error before a weighted average was taken to form
869 the final $\dndeta$. The systematic error is calculated as an average
870 in quadrature with a contribution from the residual difference between
873 Due to the nature of the ZDCvsZEM centrality determination (see
874 \cite{maxime} for details) the centrality selection of the measurement
875 with SPD, VZERO, and FMD is limited to $30\%$ central collisions. The
876 centrality bins considered are thus $0-5\%$, $5-10\%$, $10-20\%$, and
879 The selected vertices with full pseudorapidity coverage for FMD in
880 this analysis are $\cm{112.5}$, $\cm{150}$, $\cm{187.5}$, $\cm{225}$,
881 $\cm{262.5}$, $\cm{300}$. For vertices $v_z > \cm{300}$ and $v_z <
882 \cm{112.5}$ a cut is imposed in pseudorapidity to only accept data
883 with $|\eta| > 4$ to avoid regions in ALICE known to have issues with
884 the material budget description.
886 \subsection{Analysis Performance}
888 This section includes some plots to assess the validity of the
889 analysis. This includes comparisons between the measurements used
890 (SPD, VZERO, and FMD) and $\dndphi$ from the FMD.
892 Figure \ref{coverage} shows the pseudorapidity coverage of the FMD
893 when using FMD1 and FMD2I as a function of vertex with displaced
897 \includegraphics[keepaspectratio,width=\textwidth]{performance_coverage}
898 \caption{Pseudorapidity coverage of the FMD as a function of vertex
899 with displaced vertices.}
903 Figure \ref{spdfmdvzero} shows the results of the measurements of the
904 SPD, VZERO, and FMD. It is seen that there is good agreement between
905 the three different measurements albeit residual differences of up to
909 \includegraphics[keepaspectratio,width=\textwidth]{performance_spdfmdvzero}
910 \caption{$\dndeta$ measured with nominal vertices with the SPD and
911 displaced vertices with VZERO and FMD. It is seen that there is
912 good agreement between the measurements.}
916 Figure \ref{ratiofmdvzero} shows the ratios of the measurements of FMD
917 and VZERO to the combined measurement and to the SPD measurement. It
918 is seen that the residual differences are small and there is good
919 agreement between the three different measurements.
922 \begin{minipage}{0.5\linewidth}
924 \includegraphics[keepaspectratio,width=\textwidth]{performance_ratio_tocombined}
926 \begin{minipage}{0.5\linewidth}
928 \includegraphics[keepaspectratio,width=\textwidth]{performance_ratio_tospd}
930 \caption{Left: Ratio of FMD and VZERO measurements to the combined
931 $\dndeta$ measured with SPD, VZERO and FMD. Right: Ratios of FMD
932 and VZERO measurement to SPD measurement in regions of
933 overlap. It is worth pointing out that the residual differences
934 can come from the fact that the VZERO analysis uses SPD for
935 absolute calibration while the FMD analysis does not. This means that the
936 centrality determination for displaced vertices will affect the
937 FMD analysis the most because the VZERO analysis has an additional
938 constraint from the SPD analysis that uses the ZDCvsZEM centrality
939 at midrapidity where it can be crosschecked with other means of
940 centrality determination. Such a crosscheck is not possible elsewhere.}
941 \label{ratiofmdvzero}
944 Since $\dndeta$ is an average taken over $\varphi$ it is instructive
945 to consider $\dndphi$ to check that these distributions are flat as
946 they should be. Figure \ref{dndphi_pos} shows examples of the
947 $\dndphi$ distributions for FMD1. Figure \ref{dndphi_neg} shows
948 examples from FMD2 (inner ring). The two low points at $\varphi \sim
949 5.5$ in Figure \ref{dndphi_neg} are understood as coming from two
950 dying chips in FMD2I. They are considered dead in the final analysis
951 and corrected for. It is seen that the trends are quite flat within
952 $\sim 5\%$ as expected. The same trend is observed for all the
957 \includegraphics[keepaspectratio,width=\textwidth]{performance_dndphi_fmd1}
958 \caption{Examples of $\dndphi$ from FMD1 (positive
959 pseudorapidities). The distributions are essentially flat.}
965 \includegraphics[keepaspectratio,width=\textwidth]{performance_dndphi_fmd2}
966 \caption{Examples of $\dndphi$ from FMD2I (negative
967 pseudorapidities). The two low points at $\varphi \sim 5.5$ are
968 understood as the result of two dying chips in FMD2I. They are
969 considered dead in the final analysis and corrected for
970 accordingly. Apart from these points, the distributions are
975 Figure \ref{pervertex} shows the analysis performed for each
976 vertex. The material budget effects for vertices $<\cm{112.5}$ are
981 \includegraphics[keepaspectratio,width=0.8\textwidth]{performance_pervertex_negfield}
982 \includegraphics[keepaspectratio,width=0.8\textwidth]{performance_pervertex_posfield}
984 \caption{Top: Analysis per vertex for negative field data. Bottom:
985 Analysis per vertex for positive field data. In the two plots the
986 vertices where the full coverage is used are shown in blue. For
987 the red and green points there a cut is applied for the
988 pseudorapidity so that only points with $|\eta|>4$ are used in the
993 Figure \ref{leftright} shows the ratio of the postive and negative
994 pseudorapidities for the FMD. It is seen that there are discrepancies
995 of up to $\sim 5 \%$.
998 \includegraphics[keepaspectratio,width=0.7\textwidth]{performance_ratio_leftright}
999 \caption{Ratios of the positive and negative pseudorapidities for
1000 the FMD (ratio is negative over positive). The grey band indicates
1001 the combined systematic error for FMD1I and FMD2I assuming
1002 excluding all contributions from event selection and material
1003 budget (i.e. the minimum systematic error between FMD1I and
1004 FMD2I).} \label{leftright}
1007 \subsection{Results}
1009 This section summarizes the final results of the analysis and includes
1010 the figures for approval.
1012 Figure \ref{combineddndeta} shows the combined $\dndeta$ from SPD,
1013 VZERO, and FMD in the full pseudorapidity range of $-5<\eta<5.5$.
1017 \includegraphics[keepaspectratio,width=\textwidth]{results_PbPbdndeta_sat}
1018 \caption{Request for ALICE preliminary: Combined $\dndeta$ measured
1019 with SPD, VZERO and FMD. The VZERO and FMD measurements are made
1020 with displaced vertices and the SPD measurement is made at the
1021 nominal vertex. The fits are fits to a function $f(\eta) = A\exp
1022 (\frac{\eta -a_1}{2 a_2^2}) - B\exp (\frac{\eta -b_1}{2 b_{2}^2})$
1023 i.e. a Gaussian centered on $ \eta = 0$ subtracted from a similar
1025 \label{combineddndeta}
1028 Figure \ref{dndetaoverNpart} shows $dN/d\eta/(N_{part}/2)$ based on
1029 figure \ref{combineddndeta} and data taken from \cite{Aamodt:2010cz}.
1032 \includegraphics[keepaspectratio,width=\textwidth]{results_PbPbnpart_sat}
1033 \caption{Request for ALICE preliminary: The $dN/d\eta/(N_{part}/2)$
1034 measured with SPD, VZERO and FMD. The VZERO and FMD measurements
1035 are made with displaced vertices and the SPD measurement is made
1036 at the nominal vertex. The values of $N_{part}$ and the
1037 measurement at $-0.5<\eta<0.5$ taken from \cite{Aamodt:2010cz}.}
1038 \label{dndetaoverNpart}
1041 Using figure \ref{dndetaoverNpart}, figure \ref{RatiodndetaoverNpart}
1042 is constructed. It shows the ratios of $dN/d\eta/(N_{part}/2)$ in the
1043 following $\eta$ bins: $0.5<\eta<1.5$, $1.5<\eta<2.5$, $2.5<\eta<3.5$,
1044 $3.5<\eta<4.5$, and $4.5<\eta<5.5$ to the published
1045 $dN/d\eta/(N_{part}/2)$ at $-0.5<\eta<0.5$. These ratios are found to
1046 be flat for all pseudorapidity intervals.
1050 \includegraphics[keepaspectratio,width=\textwidth]{results_PbPbnpart_ratio}
1051 \caption{Request for ALICE preliminary: Ratios of
1052 $dN/d\eta/(N_{part}/2)$ at $0.5<\eta<1.5$, $1.5<\eta<2.5$,
1053 $2.5<\eta<3.5$, $3.5<\eta<4.5$, and $4.5<\eta<5.5$ to the
1054 published $dN/d\eta/(N_{part}/2)$ at $-0.5<\eta<0.5$. The ratios
1055 are found to be flat for all the pseudorapidity intervals.}
1056 \label{RatiodndetaoverNpart}
1059 With the analysis presented in figure \ref{combineddndeta} it is also
1060 possible to study longitudinal scaling from LHC to RHIC
1061 energies. Figure \ref{longscaling} shows $\dndeta$ as a function of
1062 $y'=\eta-y_{beam}$ from Figure \ref{combineddndeta} and results from
1063 the BRAHMS\cite{Bearden:2001qq} and PHOBOS\cite{Alver:2010ck}
1064 experiments at RHIC. From the figure it is seen that with the wide
1065 coverage of the SPD, VZERO, and FMD measurement it is indeed likely
1066 that longitudinal scaling exist from RHIC to LHC energies.
1070 \includegraphics[keepaspectratio,width=\textwidth]{results_PbPblongscaling_sat}
1071 \caption{Request for ALICE preliminary: Study of Longitudinal
1072 scaling. $\dndeta$ as a function of $y'=\eta-y_{beam}$ from Figure
1073 \ref{combineddndeta} and the BRAHMS\cite{Bearden:2001qq} and
1074 PHOBOS\cite{Alver:2010ck} experiments at RHIC. The fits are the
1075 function from figure \ref{combineddndeta} and a straight line
1076 ending in $\eta=y_{beam}$. From the figure it seems likely that
1077 longitudinal scaling exists from RHIC to LHC energies.}
1081 Finally the total number of produced charged particles,
1082 $N_{ch}=\int^{y_{beam}}_{-y_{beam}}\dndeta d\eta$, has
1083 been calculated from the fits in Figure \ref{combineddndeta}. The
1084 obtained values of $N_{ch}$ versus $N_{part}$ are shown in figure
1085 \ref{totalNch}. The systematic errors on $N_{ch}$ have been assessed
1086 by the procedure of varying fit functions discussed in \cite{maxime}.
1090 \includegraphics[keepaspectratio,width=\textwidth]{results_PbPbtotalnch_sat}
1091 \caption{Request for ALICE preliminary: Total number of charged
1092 particles, $N_{ch}=\int^{y_{beam}}_{-y_{beam}}\dndeta d\eta$,
1093 obtained from the fitted function in figure
1094 \ref{combineddndeta}. The systematic errors on this plot were
1095 assessed by variation of the fit function as described in \cite{maxime}.}
1099 \subsection{Comparison to old Preliminary}
1101 At QM 2011 figures were approved for preliminary status and
1102 shown. Roughly six months later it was found that the execution of the
1103 FMD analysis had a flaw\footnote{A boolean variable was wrong in a
1104 configuration macro for FMD.} which caused the results to be lower
1105 than what they should be. The top panel of Figure
1106 \ref{prelimcomparison} shows a comparison between the distribution in
1107 Figure \ref{combineddndeta} and the preliminary (ALI-PREL-2536) shown
1108 at QM 2011. The top panel shows the same comparison with the proper
1109 FMD distribution instead of the incorrect one. It is clear that the
1110 agreement observed between VZERO, SPD, and FMD at QM 2011 does not
1111 hold with the FMD analysis run properly for nominal vertices.
1115 \begin{minipage}{0.5\linewidth}
1117 \includegraphics[keepaspectratio,width=\textwidth]{oldprelim_wrong}
1119 \begin{minipage}{0.5\linewidth}
1121 \includegraphics[keepaspectratio,width=\textwidth]{oldprelim_right}
1123 \caption{Left: Comparison of new combined $\dndeta$ to the data
1124 shown at QM 2011. Right: The same comparison with the properly run
1125 FMD analysis at nominal vertices (`FMD Hits'). The difference is
1126 clearly seen around $|\eta| \sim 2$.}
1127 \label{prelimcomparison}
1130 \subsection{Summary of Systematic Errors}
1132 Table \ref{combinedsyst} shows the various sources of systematic
1133 errors for the combined measurement of VZERO, SPD, and FMD collected
1134 from Table \ref{fmdsysterror}, \cite{maxime}, and
1135 \cite{ruben,Aamodt:2010cz}. The `common' section of the table refers
1136 to source of systematic errors identified as common in the different
1137 measurements. These errors were evaluated for the displaced vertices
1138 analysis in the following way:
1140 \item Centrality errors come from variation in the parameters used in
1141 the scaling of the ZEM signal (see \cite{maxime}).
1142 \item Material budget errors were estimated by analyzing a simulation
1143 and adding a weight of $0.9$ or $1.1$ to all physical processes
1144 except decays for all secondary particles. This approach was used in
1145 the absence of suitable ALICE simulation productions.
1146 \item $p_T$ weights were developed to assess the effect of the
1147 difference in $p_T$ spectra measured by ALICE and in the HIJING
1153 \begin{tabular}{|c|c|}
1155 Source of Error & Magnitude \\
1159 Centrality & 1-4\% \\
1161 $p_T$ weights (FMD+VZERO) & 2\% \\
1163 % Strangeness Enhancement & 1\% \\
1165 Material budget(FMD+VZERO) & 4\% \\
1171 Background Subtraction & 0.1\%-2\% \\
1173 Particle Mix & 1\% \\
1175 Weak Decays & 1 \% \\
1177 Extrapolation to zero $p$ & 2\% \\
1181 Fluctuation between rings & 3\% \\
1183 Normalization & 3\%-4\% \\
1187 Variation of Cuts & 2\% \\
1189 Calculation of Multiplicity & 3\% \\
1192 \caption[Combined Systematic Errors]{The table summarizes the
1193 systematic errors in the SPD\cite{ruben,Aamodt:2010cz},
1194 VZERO\cite{maxime}, and FMD\cite{hhd:2009}.} \label{combinedsyst}
1197 The errors are obtained using variation of the quantities studied in
1198 MC simulations. In particular the studies of the dependence on the
1199 material budget are carried out with special MC simulations where the
1200 material density of ALICE is increased.
1202 \subsection{Technical Details}
1204 Here, the technical aspects of the analysis are described. The SPD
1205 analysis was done on run 137366, reconstruction pass 2 while the FMD
1206 and VZERO analysis were carried out on a total of 126 runs (46 with
1207 negative field and 80 with positive field) to obtain the necessary
1208 statistics for the displaced vertices. These runs were selected to be
1209 of good quality for VZERO, SPD, FMD, and ZDC. These data were also
1210 from pass 2 reconstruction.
1212 The AliRoot version for SPD is: \textbf{v5-03-24-AN}, for VZERO:
1213 \textbf{v5-03-28-AN}, and for FMD: \textbf{v5-03-26-AN}.
1215 For the analysis of the displaced vertices presented here the
1216 production LHC12c2 was used (the simulation was done with an anchor
1217 run for each field polarity). This production includes the latest
1218 version (as of July 2012) of the ALICE geometry and alignment.
1220 There is a twiki page for the paper using this analysis:
1221 \url{https://twiki.cern.ch/twiki/bin/viewauth/ALICE/PWGLFGeoPbPbdNdeta}.
1223 %% \currentpdfbookmark{Appendices}{Appendices}
1225 \section{Nomenclature}
1230 \begin{tabular}[t]{|lp{.8\textwidth}|}
1232 \textbf{Symbol}&\textbf{Description}\\
1234 \INEL & In--elastic event\\
1235 \INELONE & In--elastic event with at least one tracklet in the
1236 \SPD{} in the region $-1\le\eta\le1$\\
1237 \NSD{} & Non--single--diffractive event. Single diffractive
1238 events are events where one of the incident collision systems
1239 (proton or nucleus) is excited and radiates particles, but there
1240 is no other processes taking place\\
1242 $\NT{}$ & Number of events with a valid trigger\\
1243 $\NV{}$ & Number of events with a valid trigger \emph{and} a valid
1245 $\NA{}$ & Number of events with a valid trigger
1246 \emph{and} a valid vertex \emph{within} the selected vertex range.\\
1247 $\N{a,c,ac,e}{}$ & Number of events with background triggers $A$,
1248 $B$, $AC$, or $E$, \emph{and} a valid off-line trigger of the
1249 considered type. Background triggers are typically flagged with
1250 the trigger words \texttt{CINT1-A}, \texttt{CINT1-C},
1251 \texttt{CINT1-AC}, \texttt{CINT1-E}, or similar.\\
1253 $\mult{}$ & Charged particle multiplicity\\
1254 $\mult[,\text{primary}]$ & Primary charged particle multiplicity
1255 as given by simulations\\
1256 $\mult[,\text{\FMD{}}]$ & Number of charged particles that hit the
1257 \FMD{} as given by simulations\\
1258 $\mult[,t]$ & Number of charged particles in an \FMD{} strip as
1259 given by evaluating the energy response functions $F$\\
1261 $F$ & Energy response function (see \eqref{eq:energy_response})\\
1262 $\Delta_{mp}$ & Most probably energy loss\\
1263 $\xi$ & `Width' parameter of a Landau distribution\\
1264 $\sigma$ & Variance of a Gaussian distribution\\
1265 $a_i$ & Relative weight of the $i$--fold MIP peak in the energy
1268 $\Corners{}$ & Azimuthal acceptance of strip $t$\\
1269 $\SecMap{}$ & Secondary particle correction factor in $\etaphi$
1270 for a given vertex bin $v$\\
1271 $\DeadCh{}$ & Acceptance in $\etaphi$ for a given vertex bin $v$\\
1273 $\dndetadphi[incl,r,v,i]$ & Inclusive (primary \emph{and}
1274 secondary) charge particle density in event $i$ with vertex $v$,
1275 for \FMD{} ring $r$.\\
1276 $\dndetadphi[r,v,i]$ & Primary charged particle
1277 density in event $i$ with vertex $v$ for \FMD{} ring $r$. \\
1278 $\dndetadphi[v,i]$ & Primary charged particle density in event $i$
1280 $I_{v,i}(\eta)$ & $\eta$ acceptance of event $i$ with vertex $v$\\
1281 $I(\eta)$ & Integrated $\eta$ acceptance over $\NA$ events.
1282 Note, that this has a value of $\NA$ for $(\eta)$ bins where we
1283 have full coverage\\
1285 $X_t$ & Value $X$ for strip number $t$ (0-511 for inner rings,
1286 0-255 for outer rings)\\
1287 $X_r$ & Value $X$ for ring $r$ (where rings are \FMD{1i},
1288 \FMD{2i}, \FMD{2o}, \FMD{3o}, and \FMD{3i} in decreasing $\eta$
1290 $X_v$ & Value $X$ for vertex bin $v$ (typically 10 bins from -10cm
1292 $X_i$ & Value $X$ for event $i$\\
1295 \caption{Nomenclature used in this document}
1296 \label{tab:nomenclature}
1301 \section{Second pass example code}
1302 \label{app:exa_pass2}
1303 \lstset{basicstyle=\small\ttfamily,%
1304 keywordstyle=\color[rgb]{0.627,0.125,0.941}\bfseries,%
1305 identifierstyle=\color[rgb]{0.133,0.545,0.133}\itshape,%
1306 commentstyle=\color[rgb]{0.698,0.133,0.133},%
1307 stringstyle=\color[rgb]{0.737,0.561,0.561},
1308 emph={TH2D,TH1D,TFile,TTree,AliAODForwardMult},emphstyle=\color{blue},%
1309 emph={[2]dndeta,sum,norm},emphstyle={[2]\bfseries\underbar},%
1310 emph={[3]file,tree,mult,nV,nBg,nA,nT,i,gSystem},emphstyle={[3]},%
1313 \begin{lstlisting}[caption={Example 2\textsuperscript{nd} pass code to
1314 do $\dndeta$},label={lst:example},frame=single,captionpos=b]
1315 void Analyse(int mask=AliAODForwardMult::kInel,
1316 float vzLow=-10, float vzHigh=10, float trigEff=1)
1318 gSystem->Load("libANALYSIS.so"); // Load analysis libraries
1319 gSystem->Load("libANALYSISalice.so"); // General ALICE stuff
1320 gSystem->Load("libPWGLFforward2.so"); // Forward analysis code
1322 int nT = 0; // # of ev. w/trigger
1323 int nV = 0; // # of ev. w/trigger&vertex
1324 int nA = 0; // # of accepted ev.
1325 int nBg = 0; // # of background ev
1326 TH2D* sum = 0; // Summed hist
1327 AliAODForwardMult* mult = 0; // AOD object
1328 TFile* file = TFile::Open("AliAODs.root","READ");
1329 TTree* tree = static_cast<TTree*>(file->Get("aodTree"));
1330 tree->SetBranchAddress("Forward", &forward); // Set the address
1332 for (int i = 0; i < tree->GetEntries(); i++) {
1333 // Read the i'th event
1336 // Create sum histogram on first event - to match binning to input
1338 sum = static_cast<TH2D*>(mult->GetHistogram()->Clone("d2ndetadphi"));
1340 // Calculate beta=A+C-E
1341 if (mult->IsTriggerBits(mask|AliAODForwardMult::kA)) nBg++;
1342 if (mult->IsTriggerBits(mask|AliAODForwardMult::kC)) nBg++;
1343 if (mult->IsTriggerBits(mask|AliAODForwardMult::kE)) nBg--;
1345 // Other trigger/event requirements could be defined
1346 if (!mult->IsTriggerBits(mask)) continue;
1349 // Check if we have vertex and select vertex range (in centimeters)
1350 if (!mult->HasIpZ()) continue;
1353 if (!mult->InRange(vzLow, vzHigh) continue;
1356 // Add contribution from this event
1357 sum->Add(&(mult->GetHistogram()));
1360 // Get acceptance normalisation from underflow bins
1361 TH1D* norm = sum->ProjectionX("norm", 0, 0, "");
1362 // Project onto eta axis - _ignoring_underflow_bins_!
1363 TH1D* dndeta = sum->ProjectionX("dndeta", 1, -1, "e");
1364 // Normalize to the acceptance, and scale by the vertex efficiency
1365 dndeta->Divide(norm);
1366 dndeta->Scale(trigEff * nT/nV / (1 - nBg/nT), "width");
1367 // And draw the result
1372 \section{$\Delta E$ fits}
1373 \label{app:eloss_fits}
1375 \begin{figure}[htbp]
1377 \includegraphics[keepaspectratio,width=\textwidth]{eloss_fits}
1378 \caption{Summary of energy loss fits in each $\eta$ bin (see also
1379 \secref{sec:sub:sub:eloss_fits}).
1381 On the left side: Top panel shows the
1382 reduced $\chi^2$, second from the top shows the found
1383 scaling constant, 3\textsuperscript{rd} from the top is
1384 the most probable energy loss $\Delta_{mp}$, 4\textsuperscript{th}
1385 shows the width parameter $\xi$ of the Landau, and the
1386 5\textsuperscript{th} is the Gaussian width $\sigma$.
1387 $\Delta_{mp}$, $\xi$, and $\sigma$ have units of $\Delta E/\Delta
1390 On the right: The top panel shows the maximum number of
1391 multi--particle signals that where fitted, and the 4 bottom panels
1392 shows the weights $a_2,a_3,a_4,$ and $a_5$ for 2, 3, 4, and 5
1393 particle responses.}
1394 \label{fig:eloss_fits}
1398 \currentpdfbookmark{References}{References}
1399 \begin{thebibliography}{99}
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1437 %%CITATION = ARXIV:1012.1657;%%
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1443 %%CITATION = NUCL-EX/0112001;%%
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1449 %%CITATION = ARXIV:1011.1940;%%
1450 \end{thebibliography}
1454 % ispell-local-dictionary: "british"
1458 % LocalWords: tracklet diffractive IsTriggerBits AliAODForwardMult ProjectionX