1 \documentclass[11pt]{article}
2 \usepackage[margin=2cm,twoside,a4paper]{geometry}
5 \usepackage[ruled,vlined,linesnumbered]{algorithm2e}
10 \usepackage[colorlinks,urlcolor=black,hyperindex,%
11 linktocpage,a4paper,bookmarks=true,%
12 bookmarksopen=true,bookmarksopenlevel=2,%
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14 %% \usepackage{bookmark}
15 \def\AlwaysText#1{\ifmmode\relax\text{#1}\else #1\fi}
16 \newcommand{\AbbrName}[1]{\AlwaysText{{\scshape #1}}}
17 \newcommand{\CERN}{\AbbrName{cern}}
18 \newcommand{\ALICE}{\AbbrName{alice}}
19 \newcommand{\SPD}{\AbbrName{spd}}
20 \newcommand{\ESD}{\AbbrName{esd}}
21 \newcommand{\AOD}{\AbbrName{aod}}
22 \newcommand{\INEL}{\AbbrName{inel}}
23 \newcommand{\INELONE}{$\AbbrName{inel}>0$}
24 \newcommand{\NSD}{\AbbrName{nsd}}
25 \newcommand{\FMD}[1][]{\AbbrName{fmd\ifx|#1|\else#1\fi}}
26 \newcommand{\OCDB}{\AbbrName{ocdb}}
27 \newcommand{\mult}[1][]{\ensuremath N_{\text{ch}#1}}
28 \newcommand{\dndetadphi}[1][]{{\ensuremath%
29 \ifx|#1|\else\left.\fi%
30 \frac{d^2\mult{}}{d\eta\,d\varphi}%
31 \ifx|#1|\else\right|_{#1}\fi%
33 \newcommand{\landau}[1]{{\ensuremath%
34 \text{landau}\left(#1\right)}}
35 \newcommand{\dndeta}[1][]{{\ensuremath%
36 \ifx|#1|\else\left.\fi%
37 \frac{1}{N}\frac{d\mult{}}{d\eta}%
38 \ifx|#1|\else\right|_{#1}\fi%
40 \newcommand{\MC}{\AlwaysText{MC}}
41 \newcommand{\N}[2]{{\ensuremath N_{#1#2}}}
42 \newcommand{\NV}[1][]{\N{\text{V}}{#1}}
43 \newcommand{\NnotV}{\N{\not{\text{V}}}}
44 \newcommand{\NT}{\N{\text{T}}{}}
45 \newcommand{\NA}{\N{\text{A}}{}}
46 \newcommand{\Ngood}{{\ensuremath N_{\text{good}}}}
47 \newcommand{\GeV}[1]{\unit[#1]{\AlwaysText{GeV}}}
48 \newcommand{\cm}[1]{\unit[#1]{\AlwaysText{cm}}}
49 \newcommand{\secref}[1]{Section~\ref{#1}}
50 \newcommand{\figref}[1]{Figure~\ref{#1}}
51 \newcommand{\etaphi}{\ensuremath(\eta,\varphi)}
52 % Azimuthal acceptance
53 \newcommand{\Corners}{\ensuremath A^{\varphi}_{t}}
54 % Acceptance due to dead strips
55 \newcommand{\DeadCh}{\ensuremath A^{\eta}_{v,i}\etaphi}
56 \newcommand{\SecMap}{\ensuremath S_v\etaphi}
57 \setlength{\parskip}{1ex}
58 \setlength{\parindent}{0em}
59 \title{Analysing the FMD data for $\dndeta$}
60 \author{Christian Holm
61 Christensen\thanks{\texttt{$\langle$cholm@nbi.dk$\rangle$}}\quad\&\quad
62 Hans Hjersing Dalsgaard\thanks{\texttt{$\langle$canute@nbi.dk$\rangle$}}\\
63 Niels Bohr Institute\\
64 University of Copenhagen}
67 \pdfbookmark{Analysing the FMD data for dN/deta}{top}
71 \section{Introduction}
73 This document describes the steps performed in the analysis of the
74 charged particle multiplicity in the forward pseudo--rapidity
75 regions. The primary detector used for this is the \FMD{}
76 \cite{FWD:2004mz,cholm:2009}.
79 organised in 3 \emph{sub--detectors} \FMD{1}, \FMD{2}, and \FMD{3}, each
80 consisting of 1 (\FMD{1}) or 2 (\FMD{2} and~3) \emph{rings}.
81 The rings fall into two types: \emph{Inner} or \emph{outer} rings.
82 Each ring is in turn azimuthally divided into \emph{sectors}, and each
83 sector is radially divided into \emph{strips}. How many sectors,
84 strips, as well as the $\eta$ coverage is given in
85 \tablename~\ref{tab:fmd:overview}.
89 \caption{Physical dimensions of Si segments and strips.}
90 \label{tab:fmd:overview}
92 \begin{tabular}{|c|cc|cr@{\space--\space}l|r@{\space--\space}l|}
94 \textbf{Sub--detector/} &
98 \multicolumn{2}{c|}{\textbf{$r$}} &
99 \multicolumn{2}{c|}{\textbf{$\eta$}} \\
104 \multicolumn{2}{c|}{\textbf{range [cm]}} &
105 \multicolumn{2}{c|}{\textbf{coverage}} \\
107 FMD1i & 20& 512& 320 & 4.2& 17.2& 3.68& 5.03\\
108 FMD2i & 20& 512& 83.4& 4.2& 17.2& 2.28& 3.68\\
109 FMD2o & 40& 256& 75.2& 15.4& 28.4& 1.70& 2.29\\
110 FMD3i & 20& 512& -75.2& 4.2& 17.2&-2.29& -1.70\\
111 FMD3o & 40& 256& -83.4& 15.4& 28.4&-3.40& -2.01\\
117 The \FMD{} \ESD{} object contains the scaled energy deposited $\Delta
118 E/\Delta E_{mip}$ for each of the 51,200 strips. This is determined
119 in the reconstruction pass. The scaling to $\Delta E_{mip}$ is done
120 using calibration factors extracted in designated pulser runs. In
121 these runs, the front-end electronics is pulsed with an increasing
122 known pulse size, and the conversion factor from ADC counts to $\Delta
123 E_{mip}$ is determined \cite{cholm:2009}.
125 The \SPD{} is used for determination of the position of the primary
128 The analysis is performed as a two--step process.
130 \item The Event--Summary--Data (\ESD{}) is processed event--by--event
131 and passed through a number of algorithms, and
132 $\dndetadphi$ for each event is output to an Analysis--Object--Data
133 (\AOD{}) tree (see \secref{sec:gen_aod}).
134 \item The \AOD{} data is read in and the sub--sample of the data under
135 investigation is selected (e.g., \INEL{}, \INELONE{}, \NSD{}, or
136 some centrality class) and the $\dndetadphi$ histogram read in for
137 those events to build up $\dndeta$ (see \secref{sec:ana_aod}).
139 The details of each step above will be expanded upon in the
142 In Appendix~\ref{app:nomen} is an overview of the nomenclature used in
147 \section{Generating $\dndetadphi[i]$ event--by--event}
150 When reading in the \ESD{}s and generating the $\dndetadphi$
151 event--by--event the following steps are taken (in order) for each
154 \item[Event inspection] The global properties of the event is
155 determined, including the trigger type and primary interaction
156 point\footnote{`Vertex' and `primary interaction point' will be used
157 interchangeably in the text, since there is no ambiguity with
158 particle production vertex in this analysis.} $z$ coordinate (see
159 \secref{sec:sub:event_inspection}).
160 \item[Sharing filter] The \ESD{} object is read in and corrected for
161 sharing. The result is a new \ESD{} object (see
162 \secref{sec:sub:sharing_filter}).
163 \item[Density calculator] The (possibly un--corrected) \ESD{} object
164 is then inspected and an inclusive (primary \emph{and} secondary
165 particles), per--ring charged particle density
166 $\dndetadphi[incl,r,v,i]$ is made. This calculation depends in
167 general upon the interaction vertex position along the $z$ axis
168 $v_z$ (see \secref{sec:sub:density_calculator}).
169 \item[Corrections] The 5 $\dndetadphi[incl,r,v,i]$ are corrected for
170 secondary production and acceptance. The correction for the
171 secondary particle production is highly dependent on the vertex $z$
172 coordinate. The result is a per--ring, charged primary particle
173 density $\dndetadphi[r,v,i]$ (see \secref{sec:sub:corrector}).
174 \item[Histogram collector] Finally, the 5 $\dndetadphi[r,v,i]$ are
175 summed into a single $\dndetadphi[v,i]$ histogram, taking care of
176 the overlaps between the detector rings. In principle, this
177 histogram is independent of the vertex, except that the
178 pseudo--rapidity range, and possible holes in that range, depends on
179 $v_z$ --- or rather the bin in which the $v_z$ falls (see
180 \secref{sec:sub:hist_collector}).
183 Each of these steps will be detailed in the following.
185 \subsection{Event inspection}
186 \label{sec:sub:event_inspection}
188 The first thing to do, is to inspect the event for triggers. A number
189 of trigger bits, like \INEL{}, \INELONE{}, \NSD{}, and so on is then
190 propagated to the \AOD{} output.
192 Just after the sharing filter (described below) but before any further
193 processing, the vertex information is queried. If there is no vertex
194 information, or if the vertex $z$ coordinate is outside the
195 pre--defined range, then no further processing of that event takes place.
197 \subsection{Sharing filter}
198 \label{sec:sub:sharing_filter}
200 A particle originating from the vertex can, because of its incident
201 angle on the \FMD{} sensors traverse more than one strip (see
202 \figref{fig:share_fraction}). This means that the energy loss of the
203 particle is distributed over 1 or more strips. The signal in each
204 strip should therefore possibly be merged with its neighboring strip
205 signals to properly reconstruct the energy loss of a single particle.
209 \includegraphics[keepaspectratio,height=3cm]{share_fraction}
210 \caption{A particle traversing 2 strips and depositing energy in
212 \label{fig:share_fraction}
215 The effect is most pronounced in low--flux\footnote{Events with a low
216 hit density.} events, like proton--proton collisions or peripheral
217 Pb--Pb collisions, while in high--flux events the hit density is so
218 high that most likely each and every strip will be hit and the effect
219 cancel out on average.
221 Since the particles travel more or less in straight lines toward the
222 \FMD{} sensors, the sharing effect is predominantly in the $r$ or
223 \emph{strip} direction. Only neighbouring strips in a given sector is
224 therefor investigated for this effect.
226 Algorithm~\ref{algo:sharing} is applied to the signals in a given
229 \begin{algorithm}[htpb]
230 \belowpdfbookmark{Algorithm 1}{algo:sharing}
231 \SetKwData{usedThis}{current strip used}
232 \SetKwData{usedPrev}{previous strip used}
233 \SetKwData{Output}{output}
234 \SetKwData{Input}{input}
235 \SetKwData{Nstr}{\# strips}
236 \SetKwData{Signal}{current}
237 \SetKwData{Eta}{$\eta$}
238 \SetKwData{prevE}{previous strip signal}
239 \SetKwData{nextE}{next strip signal}
240 \SetKwData{lowFlux}{low flux flag}
241 \SetKwFunction{SignalInStrip}{SignalInStrip}
242 \SetKwFunction{MultiplicityOfStrip}{MultiplicityOfStrip}
243 \usedThis $\leftarrow$ false\;
244 \usedPrev $\leftarrow$ false\;
245 \For{$t\leftarrow1$ \KwTo \Nstr}{
246 \Output${}_t\leftarrow 0$\;
247 \Signal $\leftarrow$ \SignalInStrip($t$)\;
249 \uIf{\Signal is not valid}{
250 \Output${}_t \leftarrow$ invalid\;
252 \uElseIf{\Signal is 0}{
253 \Output${}_t \leftarrow$ 0\;
256 \Eta$\leftarrow$ $\eta$ of \Input${}_t$\;
257 \prevE$\leftarrow$ 0\;
258 \nextE$\leftarrow$ 0\;
260 \prevE$\leftarrow$ \SignalInStrip($t-1$)\;
263 \nextE$\leftarrow$ \SignalInStrip($t+1$)\;
265 \Output${}_t\leftarrow$
266 \MultiplicityOfStrip(\Signal,\Eta,\prevE,\nextE,\\
267 \hfill\lowFlux,$t$,\usedPrev,\usedThis)\;
270 \caption{Sharing correction}
274 Here the function \FuncSty{SignalInStrip}($t$) returns the properly
275 path--length corrected signal in strip $t$. The function
276 \FuncSty{MultiplicityOfStrip} is where the real processing takes
277 place (see page \pageref{func:MultiplicityOfStrip}).
279 \begin{function}[htbp]
280 \belowpdfbookmark{MultiplicityOfStrip}{func:MultiplicityOfStrip}
281 \caption{MultiplicityOfStrip(\DataSty{current},$\eta$,\DataSty{previous},\DataSty{next},\DataSty{low
282 flux flag},\DataSty{previous signal used},\DataSty{this signal
284 \label{func:MultiplicityOfStrip}
285 \SetKwData{Current}{current}
286 \SetKwData{Next}{next}
287 \SetKwData{Previous}{previous}
288 \SetKwData{lowFlux}{low flux flag}
289 \SetKwData{usedPrev}{previous signal used}
290 \SetKwData{usedThis}{this signal used}
291 \SetKwData{lowCut}{low cut}
292 \SetKwData{total}{Total}
293 \SetKwData{highCut}{high cut}
294 \SetKwData{Eta}{$\eta$}
295 \SetKwFunction{GetHighCut}{GetHighCut}
296 \If{\Current is very large or \Current $<$ \lowCut} {
297 \usedThis $\leftarrow$ false\;
298 \usedPrev $\leftarrow$ false\;
302 \usedThis $\leftarrow$ false\;
303 \usedPrev $\leftarrow$ true\;
306 \highCut $\leftarrow$ \GetHighCut($t$,\Eta)\;
307 %\If{\Current $<$ \Next and \Next $>$ \highCut and \lowFlux set}{
308 % \usedThis $\leftarrow$ false\;
309 % \usedPrev $\leftarrow$ false\;
312 \total $\leftarrow$ \Current\;
313 \lIf{\lowCut $<$ \Previous $<$ \highCut and not \usedPrev}{
314 \total $\leftarrow$ \total + \Previous\;
316 \If{\lowCut $<$ \Next $<$ \highCut}{
317 \total $\leftarrow$ \total + \Next\;
318 \usedThis $\leftarrow$ true\;
321 \usedPrev $\leftarrow$ true\;
324 \usedPrev $\leftarrow$ false\;
325 \usedThis $\leftarrow$ false\;
329 Here, the function \FuncSty{GetHighCut} evaluates a fit to the energy
330 distribution in the specified $\eta$ bin (see also
331 \secref{sec:sub:density_calculator}). It returns
335 where $\Delta_{mp}$ is the most probable energy loss, and $w$ is the
336 width of the Landau distribution.
338 The \KwSty{if} in line 5, says that if the previous strip was merged
339 with current one, and the signal of the current strip was added to
340 that, then the current signal is set to 0, and we mark it as used for
341 the next iteration (\DataSty{previous signal used}$\leftarrow$true).
343 % The \KwSty{if} in line 10 checks if the current signal is smaller than
344 % the next signal, if the next signal is larger than the upper cut
345 % defined above, and if we have a low--flux event\footnote{Note, that in
346 % the current implementation there are never any low--flux events.}.
347 % If that condition is met, then the current signal is the smaller of
348 % two possible candidates for merging, and it should be merged into the
349 % next signal. Note, that this \emph{only} applies in low--flux events.
352 we test if the previous signal lies between our low and
353 high cuts, and if it has not been marked as being used. If so, we add
354 it to our current signal.
356 The next \KwSty{if} on line 12 % 16
357 checks if the next signal is within our
358 cut bounds. If so, we add that signal to the current signal and mark
359 it as used for the next iteration (\DataSty{this signal
360 used}$\leftarrow$true). It will then be zero'ed on the next
361 iteration by the condition on line 6.
363 Finally, if our signal is still larger than 0, we return the signal
364 and mark this signal as used (\DataSty{previous signal
365 used}$\leftarrow$true) so that it will not be used in the next
366 iteration. Otherwise, we mark the current signal and the next signal
367 as unused and return a 0.
370 \subsection{Density calculator}
371 \label{sec:sub:density_calculator}
373 The density calculator loops over all the strip signals in the sharing
374 corrected\footnote{The sharing correction can be disabled, in which
375 case the density calculator used the input \ESD{} signals.} \ESD{}
376 and calculates the inclusive (primary + secondary) charged particle
377 density in pre--defined $\etaphi$ bins.
379 \subsubsection{Inclusive number of charged particles}
380 \label{sec:sub:sub:eloss_fits}
382 The number charged particles in a strip $\mult[,t]$ is calculated
383 using multiple Landau-like distributions fitted to the energy loss
384 spectrum of all strips in a given at a given $\eta$ bin.
386 \Delta_{i,mp} &= i (\Delta_{1,mp}+ \xi_1 \log(i))\nonumber\\
387 \xi_i &= i\xi_1\nonumber\\
388 \sigma_i &= \sqrt{i}\sigma_1\nonumber\\
389 \mult[,t] &= \frac{\sum_i^{N_{max}}
390 i\,a_i\,F(\Delta_t;\Delta_{i,mp},\xi_i,\sigma_i)}{
391 \sum_i^{N_{max}}\,a_i\,F(\Delta_t;\Delta_{i,mp},\xi_i,\sigma_i)}\quad,
393 where $F(x;\Delta_{mp},\xi,\sigma)$ is the evaluation of the Landau
394 distribution $f_L$ with most probable value $\Delta_{mp}$ and width
395 $\xi$, folded with a Gaussian distribution with spread $\sigma$ at the
396 energy loss $x$ \cite{nim:b1:16,phyrev:a28:615}.
398 \label{eq:energy_response}
399 F(x;\Delta_{mp},\xi,\sigma) = \frac{1}{\sigma \sqrt{2 \pi}}
400 \int_{-\infty}^{+\infty} d\Delta' f_{L}(x;\Delta',\xi)
401 \exp{-\frac{(\Delta_{mp}-\Delta')^2}{2\sigma^2}}\quad,
403 where $\Delta_{1,mp}$, $\xi_1$, and $\sigma_1$ are the parameters for
404 the first MIP peak, $a_1=1$, and $a_i$ is the relative weight of the
405 $i$-fold MIP peak. The parameters $\Delta_{1,mp}, \xi_1,
406 \sigma_1, \mathbf{a} = \left(a_2, \ldots a_{N_{max}}\right)$ are
409 F_j(x;C,\Delta_{mp},\xi,\sigma,\mathbf{a}) = C
410 \sum_{i=1}^{j} a_i F(x;\Delta_{i,mp},\xi_{i},\sigma_i)
412 for increasing $j$ to the energy loss spectra in separate $\eta$ bins.
413 The fit procedure is stopped when one for $j+1$
415 \item the reduced $\chi^2$ exceeds a certain threshold, or
416 \item the relative error $\delta p/p$ of any parameter of the fit
417 exceeds a certain threshold, or
418 \item when the weight $a_j+1$ is smaller than some number (typically
421 $N_{max}$ is then set to $j$. Examples of the result of these fits
422 are given in \figref{fig:eloss_fits} in Appendix~\ref{app:eloss_fits}.
424 \subsubsection{Azimuthal Acceptance}
426 Before the signal $\mult[,t]$ can be added to the $\etaphi$
427 bin in one of the 5 per--ring histograms, it needs to be corrected for
428 the $\varphi$ acceptance of the strip.
430 The sensors of the \FMD{} are not perfect arc--segments --- the two
431 top corners are cut off to allow the largest possible sensor on a 6''
432 Si-wafer. This means, however, that the strips in these outer
433 regions do not fully cover $2\pi$ in azimuth, and there is therefore a
434 need to correct for this limited acceptance.
436 The acceptance correction is only applicable where the strip length
437 does not cover the full sector. This is the case for the outer strips
438 in both the inner and outer type rings. The acceptance correction is
442 \Corners{} &= \frac{l_t}{\Delta\varphi}\quad
444 where $l_t$ is the strip length in radians at constant $r$, and
445 $\Delta\varphi$ is $2\pi$ divided by the number of sectors in the
446 ring (20 for inner type rings, and 40 for outer type rings).
448 Note, that this correction is a hardware--related correction, and does
449 not depend on the properties of the collision (e.g., primary vertex
452 The final $\etaphi$ content of the 5 output vertex dependent,
453 per--ring histograms of the inclusive charged particle density is then
457 \dndetadphi[incl,r,v,i\etaphi] &= \sum_t^{t\in\etaphi}
458 \mult[,t]\,\Corners{}
460 where $t$ runs over the strips in the $\etaphi$ bin.
462 \subsection{Corrections}
463 \label{sec:sub:corrector}
465 The corrections code receives the five vertex dependent,
466 per--ring histograms of the inclusive charged particle density
467 $\dndetadphi[incl,r,v,i]$ from the density calculator and applies
470 \subsubsection{Secondary correction}
472 %% hHits_FMD<d><r>_vtx<v>
473 %% hCorrection = -----------------------
474 %% hPrimary_FMD_<r>_vtx<v>
477 %% - hPrimary_FMD_<r>_vtx<vtx> is 2D of eta,phi for all primary ch
479 %% - hHits_FMD<d><r>_vtx<v> is 2D of eta,phi for all track-refs that
480 %% hit the FMD - The 2D version of hMCHits_nocuts_FMD<d><r>_vtx<v>
482 This is a 2 dimensional histogram generated from simulations, as the
483 ratio of primary particles to the total number of particles that fall
484 within an $\etaphi$ bin for a given vertex bin
489 \frac{\sum_i^{\NV[,v]}\mult[,\text{primary},i]\etaphi}{
490 \sum_i^{\NV[,v]}\mult[,\text{\FMD{}},i]\etaphi}\quad,
492 where $\NV[,v]$ is the number of events with a valid trigger and a
493 vertex in bin $v$, and $\mult[,\FMD{},i]$ is the total number of
494 charged particles that hit the \FMD{} in event $i$ in the specified
495 $\etaphi$ bin and $\mult[,\text{primary},i]$ is number of
496 primary charged particles in event $i$ within the specified
499 $\mult[,\text{primary}]\etaphi$ is given by summing over the
500 charged particles labelled as primaries \emph{at the time of the
501 collision} as defined in the simulation code. That is, it is the
502 number of primaries within the $\etaphi$ bin at the collision
503 point --- not at the \FMD{}.
505 $\SecMap$ is varies from $\approx 1.5$ for the most forward bins to
506 $\approx 3$ for the more central bins. For pp, different event
507 generators were used and found to give compatible results within
508 3--5\%. For pp, at least some millions of events must be
509 accumulated to reach satisfactory statistics. For Pb--Pb where the
510 general hit density is larger, reasonable statistics can be achieved
513 \subsubsection{Acceptance due to dead channels}
515 Some of the strips in the \FMD{} have been marked up as \emph{dead},
516 meaning that they are not used in the reconstruction or analysis.
517 This leaves holes in the acceptance of each defined $\etaphi$
518 which need to be corrected for.
520 Dead channels are marked specially in the \ESD{}s with the flag
521 \textit{Invalid Multiplicity}. This is used in the analysis to build
522 up and event--by--event acceptance correction in each $\etaphi$
523 bin by calculating the ratio
525 \label{eq:dead_channels}
527 \frac{\sum_t^{t\in\etaphi}\left\{\begin{array}{cl}
528 1 & \text{if not dead}\\
530 \end{array}\right.}{\sum_t^{t\in\etaphi} 1}\quad,
532 where $t$ runs over the strips in the $\etaphi$ bin. This correction
533 is obviously $v_z$ dependent since which $\etaphi$ bin a strip $t$
534 corresponds to depends on its relative position to the primary vertex.
536 Alternatively, pre--made acceptance factors can be used. These are
537 made from the off-line conditions database (\OCDB{}).
539 The 5 output vertex dependent, per--ring histograms of the primary
540 charged particle density is then given by
542 \dndetadphi[r,v,i\etaphi] &=
543 \SecMap{} \frac{1}{\DeadCh{}}\dndetadphi[incl,r,v,i\etaphi]
546 \subsection{Histogram collector}
547 \label{sec:sub:hist_collector}
549 The histogram collector collects the information from the 5 vertex
550 dependent, per--ring histograms of the primary charged particle
551 density $\dndetadphi[r,v,i]$ into a single vertex dependent histogram
552 of the charged particle density $\dndetadphi[v,i]$.
554 To do this, it first calculates, for each vertex bin, the $\eta$ bin
555 range to use for each ring. It investigates the secondary correction
556 maps $\SecMap{}$ to find the edges of each map. The edges are given
557 by the $\eta$ range where $\SecMap{}$ is larger than some
558 threshold\footnote{Typically $t_s\approx 0.1$.} $t_s$. The code
559 applies safety margin of a $N_{cut}$ bins\footnote{Typically
560 $N_{cut}=1$.}, to ensure that the data selected does not have too
561 large corrections associated with it.
563 It then loops over the bins in the defined $\eta$ range and sums the
564 contributions from each of the 5 histograms. In the $\eta$ ranges
565 where two rings overlap, the collector calculates the average and adds
566 the errors in quadrature\footnote{While not explicitly checked, it was
567 found that the histograms agrees within error bars in the
570 The output vertex dependent histogram of the primary
571 charged particle density is then given by
574 \dndetadphi[v,i\etaphi] &=
575 \frac{1}{N_{r\in\etaphi}}\sum_{r}^{r\in\etaphi}
576 \dndetadphi[r,v,i\etaphi]\\
577 \delta\left[\dndetadphi[v,i\etaphi]\right] &=
578 \frac{1}{N_{r\in\etaphi}}\sqrt{\sum_{r}^{r\in\etaphi}
579 \delta\left[\dndetadphi[r,v,i\etaphi]\right]^2}
582 where $N_{r\in\etaphi}$ is the number of overlapping histograms
583 in the given $\etaphi$ bin.
585 The histogram collector stores the found $\eta$ ranges in the
586 underflow bin of the histogram produced. The content of the overflow
591 \frac{1}{N_{r\in(\eta)}}
592 \sum_{r}^{r\in(\eta)} \left\{\begin{array}{cl}
593 0 & \eta \text{\ bin not selected}\\
594 1 & \eta \text{\ bin selected}
595 \end{array}\right.\quad,
597 where $N_{r\in(\eta)}$ is the number of overlapping histograms in the
598 given $\eta$ bin. The subscript $v$ indicates that the content
599 depends on the current vertex bin of event $i$.
601 \section{Building the final $\dndeta$}
604 To build the final $\dndeta$ distribution it is enough to sum
605 \eqref{eq:superhist} and \eqref{eq:overflow} over all interesting
606 events and correct for the acceptance $I(\eta)$
608 \dndetadphi[\etaphi] &= \sum_i^{\NA}\dndetadphi[i,v\etaphi]\\
609 I(\eta) &= \sum_i^{\NA}I_{i,v}(\eta)\quad.
611 Note, that $I(\eta)\le\NA$.
613 We then need to normalise to the total number of events $N_X$, given
616 \N{X}{} &= \frac{1}{\epsilon_X}\left[\NA + \alpha(\NnotV -
617 \beta)\right] \label{eq:fulleventnorm}\\
618 & = \frac{1}{\epsilon_X}\left[\NA + \frac{\NA}{\NV}(\NT-\NV{} -
619 \beta)\right]\nonumber \\
620 & =\frac{1}{\epsilon_X}\NA\left[1+\frac{1}{\epsilon_V}-1-
621 \frac{\beta}{\NV}\right]\nonumber\\
622 & = \frac{1}{\epsilon_X}\frac{1}{\epsilon_V}\NA
623 \left(1-\frac{\beta}{\NT{}}\right)\nonumber
627 \item[$\epsilon_X$] is the trigger efficiency for type
628 $X\in[\text{\INEL},\text{\INELONE},\text{\NSD},...]$
629 \item[$\epsilon_V=\frac{\NV{}}{\NT{}}$] is the vertex efficiency
630 evaluated over the data.
631 \item[$\NA$] is the number of events with a trigger \emph{and} a valid
632 vertex in the selected range
633 \item[$\NV{}$] is the number of events with a trigger \emph{and} a valid
635 \item[$\NT$] is the number of events with a trigger.
636 \item[$\NnotV{}=\NT-\NV{}$] is the number of events with a trigger
637 \emph{but no} valid vertex
638 \item[$\alpha=\frac{\NA}{\NV}$] is the fraction of accepted events of
639 the total number of events with a trigger and valid vertex.
640 \item[$\beta=\N{a}{}+\N{b}{}-\N{e}{}$] is the number of background
641 events \emph{with} a valid off-line trigger.
643 The two terms under the parenthesis in \eqref{eq:fulleventnorm} refers
644 to the observed number of event $\NA$, and the events missed because
645 of no vertex reconstruction. Note, for $\beta\ll\NT{}$
646 \eqref{eq:fulleventnorm} reduces to the simpler expression
648 \N{X}{} = \frac1{\epsilon_X}\frac1{\epsilon_V}\NA{}
650 The trigger efficiency $\epsilon_X$ for a given trigger type $X$ is
651 evaluated from simulations as
653 \epsilon_X = \frac{\N{X\wedge \text{T}}{}}{\N{X}{}}\quad,
655 that is, the ratio of number of events of type $X$ with a
656 corresponding trigger to the number of events of type $X$.
658 The final event--normalised charged particle density then becomes
660 \frac{1}{N}\frac{dN_{\text{ch}}}{d\eta} &=
661 \frac{1}{\N{X}{}} \int_0^{2\pi} d\varphi
662 \frac{\dndetadphi[\etaphi]}{I(\eta)}
663 \label{eq:eventnormdndeta}
666 If the trigger $X$ introduces a bias on the measured number of events,
667 then \eqref{eq:eventnormdndeta} need to be modified to
669 \frac{1}{N}\frac{dN_{\text{ch}}}{d\eta} &=
670 \frac{1}{\N{X}{}} \int_0^{2\pi} d\varphi
671 \frac{\frac{1}{B\etaphi}\dndetadphi[\etaphi]}{I(\eta)}
672 \label{eq:eventnormdndeta2}\quad,
674 where $B\etaphi$ is the bias correction. This is typically
675 calculated from simulations using the expression
677 B\etaphi = \frac{\frac{1}{\N{X\wedge
678 \text{T}}{}}\sum_i^{\N{X\wedge \text{T}}{}}
679 \mult[,\text{primary}]\etaphi}{\frac{1}{\N{X}{}}\sum_i^{\N{X}{}}
680 \mult[,\text{primary}]\etaphi}
684 \section{Using the per--event $\dndetadphi[i,v]$ histogram for other
687 \subsection{Multiplicity distribution}
689 To build the multiplicity distribution for a given $\eta$ range
690 $[\eta_1,\eta_2]$, one needs to find the total multiplicity in that
691 $\eta$ range for each event. To do so, one should sum the
692 $\dndetadphi[i,v]$ histogram over all $\varphi$ and in the selected
695 n'_{i[\eta_1,\eta_2]}, &= \int_{\eta_1}^{\eta_2}d\eta\int_0^{2\pi}d\varphi
696 \dndetadphi[i,v]\quad.\nonumber
698 However, $n'_i$ is not corrected for the coverage in $\eta$ for the
699 particular vertex range $v$. One therefor needs to correct for the
700 number of missing bins in the range $[\eta_1,\eta_2]$. Suppose
701 $[\eta_1,\eta_2]$ covers $N_{[\eta_1,\eta_2]}$ $\eta$ bins, then the acceptance
702 correction is given by
704 A_{i,[\eta_1,\eta_2]} = \frac{N_{[\eta_1,\eta_2]}}{\int_{\eta_1}^{\eta_2}d\eta\,
705 I_{i,v}(\eta)}\quad.\nonumber
707 The per--event multiplicity is then given by
709 n_{i,[\eta_1,\eta_2]} &= A_{i,[\eta_1,\eta_2]}\,n'_{i,[\eta_1,\eta_2]}\nonumber\\
710 &= \frac{N_{[\eta_1,\eta_2]}}{\int_{\eta_1}^{\eta_2}\eta
711 I_{i,v}(\eta)} \int_{\eta_1}^{\eta_2}d\eta\int_0^{2\pi}d\varphi
716 \subsection{Forward--Backward correlations}
718 To do forward--backward correlations, one need to calculate
719 $n_{i,[\eta_1,\eta_2]}$ as shown in \eqref{eq:event_n} in two bins
720 $n_{i,[\eta_1,\eta_2]}$ and $n_{i,[-\eta_2,-\eta_1]}$ \textit{e.g.},
721 $n_{i,f}=n_{i,[-3,-1]}$ and $n_{i,b}=n_{i,[1,3]}$.
724 \section{Some results}
726 %% \figurename{}s \ref{fig:1} to \ref{fig:3} shows some results.
727 Figures below show some examples. Note these are not finalised
732 \includegraphics[keepaspectratio,width=\textwidth]{%
733 dndeta_pp_0900GeV_INEL_m10p10cm}
734 \caption{$\dndeta$ for pp for \INEL{} events at $\sqrt{s}=\GeV{900}$,
735 $\cm{-10}\le v_z\le\cm{10}$, rebinned by a factor 5. Middle panel
736 shows the ratio of ALICE data to UA5, and the bottom panel shows
737 the ratio of the right (positive) side to the left (negative) side
738 of the forward $\dndeta$.}
745 \includegraphics[keepaspectratio,width=\textwidth]{%
746 dndeta_0900GeV_m10-p10cm_rb05_inelgt0}
747 \caption{$\dndeta$ for pp for \INELONE{} events at
748 $\sqrt{s}=\GeV{900}$, $\cm{-10}\le v_z\le\cm{10}$, rebinned by a
749 factor 5. Comparisons to other measurements shown where
755 \includegraphics[keepaspectratio,width=\textwidth]{%
756 dndeta_0900GeV_m10-p10cm_rb05_nsd}
757 \caption{$\dndeta$ for pp for \NSD{} events at $\sqrt{s}=\GeV{900}$,
758 $\cm{-10}\le v_z\le\cm{10}$, rebinned by a factor 5. Comparisons
759 to other measurements shown where applicable}
765 %% \currentpdfbookmark{Appendices}{Appendices}
767 \section{Nomenclature}
772 \begin{tabular}[t]{|lp{.8\textwidth}|}
774 \textbf{Symbol}&\textbf{Description}\\
776 \INEL & In--elastic event\\
777 \INELONE & In--elastic event with at least one tracklet in the
778 \SPD{} in the region $-1\le\eta\le1$\\
779 \NSD{} & Non--single--diffractive event. Single diffractive
780 events are events where one of the incident collision systems
781 (proton or nucleus) is excited and radiates particles, but there
782 is no other processes taking place\\
784 $\NT{}$ & Number of events with a valid trigger\\
785 $\NV{}$ & Number of events with a valid trigger \emph{and} a valid
787 $\NA{}$ & Number of events with a valid trigger
788 \emph{and} a valid vertex \emph{within} the selected vertex range.\\
789 $\N{a,c,ac,e}{}$ & Number of events with background triggers $A$,
790 $B$, $AC$, or $E$, \emph{and} a valid off-line trigger of the
791 considered type. Background triggers are typically flagged with
792 the trigger words \texttt{CINT1-A}, \texttt{CINT1-C},
793 \texttt{CINT1-AC}, \texttt{CINT1-E}, or similar.\\
795 $\mult{}$ & Charged particle multiplicity\\
796 $\mult[,\text{primary}]$ & Primary charged particle multiplicity
797 as given by simulations\\
798 $\mult[,\text{\FMD{}}]$ & Number of charged particles that hit the
799 \FMD{} as given by simulations\\
800 $\mult[,t]$ & Number of charged particles in an \FMD{} strip as
801 given by evaluating the energy response functions $F$\\
803 $F$ & Energy response function (see \eqref{eq:energy_response})\\
804 $\Delta_{mp}$ & Most probably energy loss\\
805 $\xi$ & `Width' parameter of a Landau distribution\\
806 $\sigma$ & Variance of a Gaussian distribution\\
807 $a_i$ & Relative weight of the $i$--fold MIP peak in the energy
810 $\Corners{}$ & Azimuthal acceptance of strip $t$\\
811 $\SecMap{}$ & Secondary particle correction factor in $\etaphi$
812 for a given vertex bin $v$\\
813 $\DeadCh{}$ & Acceptance in $\etaphi$ for a given vertex bin $v$\\
815 $\dndetadphi[incl,r,v,i]$ & Inclusive (primary \emph{and}
816 secondary) charge particle density in event $i$ with vertex $v$,
817 for \FMD{} ring $r$.\\
818 $\dndetadphi[r,v,i]$ & Primary charged particle
819 density in event $i$ with vertex $v$ for \FMD{} ring $r$. \\
820 $\dndetadphi[v,i]$ & Primary charged particle density in event $i$
822 $I_{v,i}(\eta)$ & $\eta$ acceptance of event $i$ with vertex $v$\\
823 $I(\eta)$ & Integrated $\eta$ acceptance over $\NA$ events.
824 Note, that this has a value of $\NA$ for $(\eta)$ bins where we
827 $X_t$ & Value $X$ for strip number $t$ (0-511 for inner rings,
828 0-255 for outer rings)\\
829 $X_r$ & Value $X$ for ring $r$ (where rings are \FMD{1i},
830 \FMD{2i}, \FMD{2o}, \FMD{3o}, and \FMD{3i} in decreasing $\eta$
832 $X_v$ & Value $X$ for vertex bin $v$ (typically 10 bins from -10cm
834 $X_i$ & Value $X$ for event $i$\\
837 \caption{Nomenclature used in this document}
838 \label{tab:nomenclature}
843 \section{Second pass example code}
844 \label{app:exa_pass2}
845 \lstset{basicstyle=\small\ttfamily,%
846 keywordstyle=\color[rgb]{0.627,0.125,0.941}\bfseries,%
847 identifierstyle=\color[rgb]{0.133,0.545,0.133}\itshape,%
848 commentstyle=\color[rgb]{0.698,0.133,0.133},%
849 stringstyle=\color[rgb]{0.737,0.561,0.561},
850 emph={TH2D,TH1D,TFile,TTree,AliAODForwardMult},emphstyle=\color{blue},%
851 emph={[2]dndeta,sum,norm},emphstyle={[2]\bfseries\underbar},%
852 emph={[3]file,tree,mult,nV,nBg,nA,nT,i,gSystem},emphstyle={[3]},%
855 \begin{lstlisting}[caption={Example 2\textsuperscript{nd} pass code to
856 do $\dndeta$},label={lst:example},frame=single,captionpos=b]
857 void Analyse(int mask=AliAODForwardMult::kInel,
858 float vzLow=-10, float vzHigh=10, float trigEff=1)
860 gSystem->Load("libANALYSIS.so"); // Load analysis libraries
861 gSystem->Load("libANALYSISalice.so"); // General ALICE stuff
862 gSystem->Load("libPWGLFforward2.so"); // Forward analysis code
864 int nT = 0; // # of ev. w/trigger
865 int nV = 0; // # of ev. w/trigger&vertex
866 int nA = 0; // # of accepted ev.
867 int nBg = 0; // # of background ev
868 TH2D* sum = 0; // Summed hist
869 AliAODForwardMult* mult = 0; // AOD object
870 TFile* file = TFile::Open("AliAODs.root","READ");
871 TTree* tree = static_cast<TTree*>(file->Get("aodTree"));
872 tree->SetBranchAddress("Forward", &forward); // Set the address
874 for (int i = 0; i < tree->GetEntries(); i++) {
875 // Read the i'th event
878 // Create sum histogram on first event - to match binning to input
880 sum = static_cast<TH2D*>(mult->GetHistogram()->Clone("d2ndetadphi"));
882 // Calculate beta=A+C-E
883 if (mult->IsTriggerBits(mask|AliAODForwardMult::kA)) nBg++;
884 if (mult->IsTriggerBits(mask|AliAODForwardMult::kC)) nBg++;
885 if (mult->IsTriggerBits(mask|AliAODForwardMult::kE)) nBg--;
887 // Other trigger/event requirements could be defined
888 if (!mult->IsTriggerBits(mask)) continue;
891 // Check if we have vertex and select vertex range (in centimeters)
892 if (!mult->HasIpZ()) continue;
895 if (!mult->InRange(vzLow, vzHigh) continue;
898 // Add contribution from this event
899 sum->Add(&(mult->GetHistogram()));
902 // Get acceptance normalisation from underflow bins
903 TH1D* norm = sum->ProjectionX("norm", 0, 0, "");
904 // Project onto eta axis - _ignoring_underflow_bins_!
905 TH1D* dndeta = sum->ProjectionX("dndeta", 1, -1, "e");
906 // Normalize to the acceptance, and scale by the vertex efficiency
907 dndeta->Divide(norm);
908 dndeta->Scale(trigEff * nT/nV / (1 - nBg/nT), "width");
909 // And draw the result
914 \section{$\Delta E$ fits}
915 \label{app:eloss_fits}
919 \includegraphics[keepaspectratio,width=\textwidth]{eloss_fits}
920 \caption{Summary of energy loss fits in each $\eta$ bin (see also
921 \secref{sec:sub:sub:eloss_fits}).
923 On the left side: Top panel shows the
924 reduced $\chi^2$, second from the top shows the found
925 scaling constant, 3\textsuperscript{rd} from the top is
926 the most probable energy loss $\Delta_{mp}$, 4\textsuperscript{th}
927 shows the width parameter $\xi$ of the Landau, and the
928 5\textsuperscript{th} is the Gaussian width $\sigma$.
929 $\Delta_{mp}$, $\xi$, and $\sigma$ have units of $\Delta E/\Delta
932 On the right: The top panel shows the maximum number of
933 multi--particle signals that where fitted, and the 4 bottom panels
934 shows the weights $a_2,a_3,a_4,$ and $a_5$ for 2, 3, 4, and 5
936 \label{fig:eloss_fits}
940 \currentpdfbookmark{References}{References}
941 \begin{thebibliography}{99}
942 \bibitem{FWD:2004mz} \ALICE{} Collaboration, Bearden, I.~G.\ \textit{et al}
943 \textit{ALICE technical design report on forward detectors: FMD, T0
944 and V0}, \CERN{}, 2004, CERN-LHCC-2004-025
945 \bibitem{cholm:2009} Christensen, C.~H., \textit{The ALICE Forward
946 Multiplicity Detector --- From Design to Installation},
947 Ph.D.~thesis, University of Copenhagen, 2009,
948 \url{http://www.nbi.dk/~cholm/}.
950 %% \bibitem{Hancock:1983ry}
951 S.~Hancock, F.~James, J.~Movchet {\it et al.},
952 ``Energy Loss Distributions For Single Particles And Several
953 Particles In A Thin Silicon Absorber,'' Nucl.\ Instrum.\ Meth.\
954 \textbf{B1} (1984) 16, \url{http://cdsweb.cern.ch/record/147286/files/cer-000058451.pdf}.
955 \bibitem{phyrev:a28:615}
956 %% \bibitem{Hancock:1983fp}
957 S.~Hancock, F.~James, J.~Movchet {\it et al.}, ``Energy Loss And
958 Energy Straggling Of Protons And Pions In The Momentum Range
959 0.7-gev/c To 115-gev/c,'' Phys.\ Rev.\ \textbf{A28} (1983) 615,
960 \url{http://cdsweb.cern.ch/record/145395/files/PhysRevA.28.615.pdf}.
961 \end{thebibliography}
965 % ispell-local-dictionary: "british"
968 % LocalWords: tracklet diffractive IsTriggerBits AliAODForwardMult ProjectionX