1 \documentclass[11pt]{article}
2 \usepackage[margin=2cm,twoside,a4paper]{geometry}
5 \usepackage[ruled,vlined,linesnumbered]{algorithm2e}
6 \newcommand{\AbbrName}[1]{{\ifmmode\text{\scshape #1}\else{\scshape #1}\fi}}
7 \newcommand{\SPD}{\AbbrName{spd}}
8 \newcommand{\ESD}{\AbbrName{esd}}
9 \newcommand{\AOD}{\AbbrName{aod}}
10 \newcommand{\INEL}{\AbbrName{inel}}
11 \newcommand{\INELONE}{$\AbbrName{inel}>0$}
12 \newcommand{\NSD}{\AbbrName{nsd}}
13 \newcommand{\FMD}[1][]{\AbbrName{nsd\ifx|#1|\else#1\fi}}
14 \newcommand{\dndetadphi}[1][]{{\ensuremath%
15 \ifx|#1|\else\left.\fi%
16 \frac{d^2N_{ch}}{d\eta\,d\varphi}%
17 \ifx|#1|\else\right|_{#1}\fi%
19 \newcommand{\landau}[1]{{\ensuremath%
20 \text{landau}\left(#1\right)}}
21 \newcommand{\dndeta}[1][]{{\ensuremath%
22 \ifx|#1|\else\left.\fi%
23 \frac{1}{N}\frac{dN_{ch}}{d\eta}%
24 \ifx|#1|\else\right|_{#1}\fi%
26 \setlength{\parskip}{1ex}
27 \setlength{\parindent}{0em}
29 \title{Analysing the FMD data for $\dndeta$}
30 \author{Christian Holm
31 Christensen\footnote{\texttt{$\langle$cholm@nbi,dk$\rangle$}}\quad\&\quad
32 Hans Hjersing Dalsgaard\footnote{\texttt{$\langle$canute@nbi,dk$\rangle$}}\\
33 Niels Bohr Institute\\
34 University of Copenhagen}
39 \section*{Introduction}
41 This document describes the steps performed in the analysis of the
42 charged particle multiplicity in the forward pseudo--rapidity
45 The analysis is performed as a two--step process.
47 \item The Event--Summary--Data (\ESD{}) is processed event--by--event
48 and passed through a number of algorithms, and
49 $\dndetadphi$ for each event is output to an Analysis--Object--Data
51 \item The \AOD{} data is read in and the sub--sample of the data under
52 investigation is selected (e.g., \INEL{}, \INELONE{}, \NSD{}, or
53 some centrality class) and the $\dndetadphi$ histogram read in for
54 those events to build up $\dndeta$
56 The details of each step above will be expanded upon in the
59 \section*{Generating $\dndetadphi[i]$ event--by--event}
61 When reading in the \ESD{}s and generating the $\dndetadphi$
62 event--by--event the following steps are taken (in order) for each
65 \item[Event inspection] The global properties of the event is
66 determined, including the trigger type, vertex $z$ coordinate, and
67 whether this is a low--flux event or not.
68 \item[Sharing filter] The \ESD{} object is read in and corrected for
69 sharing. The result is a new \ESD{} object.
70 \item[Density calculator] The (possibly un--corrected) \ESD{} object
71 is then inspected and an inclusive, per--ring charged particle
72 density $\dndetadphi[incl,r,v,i]$ is made. This
73 calculation depends in general upon the interaction
74 vertex\footnote{In the following simply labelled 'primary vertex' or
75 'vertex'.} position along the $z$ axis ($v_z$).
76 \item[Corrections] The 5 $\dndetadphi[incl,r,v,i]$ are
77 corrected for secondary production, event selection efficiency, and
78 possibly the sharing efficiency. These corrections are highly
79 dependent on the vertex $z$ coordinate. The result is an per--ring,
80 charged primary particle density $\dndetadphi[r,v,i]$
81 \item[Histogram collector] Finally, the 5 $\dndetadphi[r,v,i]$ are
82 summed into a single $\dndetadphi[v,i]$ histogram, taking care of
83 the overlaps between the detector rings. In principle, this
84 histogram is independent of the vertex, except that the
85 pseudo--rapidity range, and possible holes in that range, depends on
86 $v_z$ --- or rather the bin in which the $v_z$ falls.
89 Each of these steps will be detailed in the following.
91 \subsection*{Event inspection}
93 The first thing to do, is to inspect the event for triggers. A number
94 of trigger bits, like \INEL{}, \INELONE{}, \NSD{}, and so on is then
95 propagated to the \AOD{} output.
97 Next, the number of \emph{tracklets} reconstructed in the Silicon
98 Pixel Detector (\SPD{}) compared to a threshold. If the number of
99 track--lets falls belows this threshold, the event is consider a
102 Just after the sharing filter (described below) but before any more
103 processing, the vertex information is queried. If there is no vertex
104 information, or if the vertex $z$ coordinate is outside the
105 pre--defined range, then no further processing takes place.
107 \subsection*{Sharing filter}
109 The \FMD{} \ESD{} object contains the scaled energy deposited $\Delta
110 E/\Delta E_{mip}$ for each of the 51,200 strips. The \FMD{} is
111 organised in 3 \emph{sub--detectors} \FMD{1}, \FMD{2}, and \FMD{3}, each
112 consisting of 1 (\FMD{1}) or 2 (\FMD{2} and \FMD{3}) \emph{rings}.
113 The rings fall into two types: \emph{Inner} or \emph{outer} rings.
114 Each ring is in turn azimuthal divided into \emph{sectors}, and each
115 sector is radially divided into \emph{strips}. How many sectors,
116 strips, as well as the $\eta$ coverage is given in
117 \tablename~\ref{tab:fmd:overview}.
121 \caption{Physical dimensions of Si segments and strips.}
122 \label{tab:fmd:overview}
124 \begin{tabular}{|c|cc|cr@{\space--\space}l|r@{\space--\space}l|}
126 \textbf{Sub--detector/} &
130 \multicolumn{2}{c|}{\textbf{$r$}} &
131 \multicolumn{2}{c|}{\textbf{$\eta$}} \\
136 \multicolumn{2}{c|}{\textbf{range [cm]}} &
137 \multicolumn{2}{c|}{\textbf{coverage}} \\
139 FMD1i & 20& 512& 320 & 4.2& 17.2& 3.68& 5.03\\
140 FMD2i & 20& 512& 83.4& 4.2& 17.2& 2.28& 3.68\\
141 FMD2o & 40& 256& 75.2& 15.4& 28.4& 1.70& 2.29\\
142 FMD3i & 20& 512& -75.2& 4.2& 17.2&-2.29& -1.70\\
143 FMD3o & 40& 256& -83.4& 15.4& 28.4&-3.40& -2.01\\
149 A particle originating from the vertex can, because of it's incident
150 angle on the \FMD{} sensors traverse more than one strip. That means
151 that the energy loss of the particle is distributed over 1 or more
152 strips. The signal in each strip should therefore possibly merged
153 with it's neighbor strip signals to properly reconstruct the energy
154 loss of a single particle.
156 The effect is most pronounced in low--flux events, like proton--proton
157 collisions or peripheral Pb--Pb collisions, while in high--flux events
158 the hit density is so high that most likely each and every strip will
159 be hit and the effect cancel out on average.
161 Since the particles travel more or less in straight lines toward the
162 \FMD{} sensors, the sharing effect predominantly in the $r$ or
163 \emph{strip} direction. Only neighboring strips in a given sector is
164 therefor investigated for this effect.
166 Algorithm~\ref{algo:sharing} is applied to the signals in a given
169 \begin{algorithm}[htpb]
170 \SetKwData{usedThis}{current strip used}
171 \SetKwData{usedPrev}{previous strip used}
172 \SetKwData{Output}{output}
173 \SetKwData{Input}{input}
174 \SetKwData{Nstr}{\# strips}
175 \SetKwData{Signal}{current}
176 \SetKwData{Eta}{$\eta$}
177 \SetKwData{prevE}{previous strip signal}
178 \SetKwData{nextE}{next strip signal}
179 \SetKwData{lowFlux}{low flux flag}
180 \SetKwFunction{SignalInStrip}{SignalInStrip}
181 \SetKwFunction{MultiplicityOfStrip}{MultiplicityOfStrip}
182 \usedThis $\leftarrow$ false\;
183 \usedPrev $\leftarrow$ false\;
184 \For{$t\leftarrow1$ \KwTo \Nstr}{
185 \Output${}_t\leftarrow 0$\;
186 \Signal $\leftarrow$ \SignalInStrip($t$)\;
188 \uIf{\Signal is not valid}{
189 \Output${}_t \leftarrow$ invalid\;
191 \uElseIf{\Signal is 0}{
192 \Output${}_t \leftarrow$ 0\;
195 \Eta$\leftarrow$ $\eta$ of \Input${}_t$\;
196 \prevE$\leftarrow$ 0\;
197 \nextE$\leftarrow$ 0\;
199 \prevE$\leftarrow$ \SignalInStrip($t-1$)\;
202 \nextE$\leftarrow$ \SignalInStrip($t+1$)\;
204 \Output${}_t\leftarrow$
205 \MultiplicityOfStrip(\Signal,\Eta,\prevE,\nextE,\\
206 \hfill\lowFlux,$t$,\usedPrev,\usedThis)\;
209 \caption{Sharing correction}
213 Here the function \FuncSty{SignalInStrip}($t$) returns the properly
214 path--length corrected signal in strip $t$. The function
215 \FuncSty{MultiplicityInStrip} is where the real processing takes
216 place (see page \pageref{func:MultiplicityInStrip}).
218 \begin{function}[htbp]
219 \caption{MultiplicityInStrip(\DataSty{current},$\eta$,\DataSty{previous},\DataSty{next},\DataSty{low
220 flux flag},\DataSty{previous signal used},\DataSty{this signal
222 \label{func:MultiplicityInStrip}
223 \SetKwData{Current}{current}
224 \SetKwData{Next}{next}
225 \SetKwData{Previous}{previous}
226 \SetKwData{lowFlux}{low flux flag}
227 \SetKwData{usedPrev}{previous signal used}
228 \SetKwData{usedThis}{this signal used}
229 \SetKwData{lowCut}{low cut}
230 \SetKwData{total}{Total}
231 \SetKwData{highCut}{high cut}
232 \SetKwData{Eta}{$\eta$}
233 \SetKwFunction{GetHighCut}{GetHighCut}
234 \If{\Current is very large or \Current $<$ \lowCut} {
235 \usedThis $\leftarrow$ false\;
236 \usedPrev $\leftarrow$ false\;
240 \usedThis $\leftarrow$ false\;
241 \usedPrev $\leftarrow$ true\;
244 \highCut $\leftarrow$ \GetHighCut($t$,\Eta)\;
245 \If{\Current $<$ \Next and \Next $>$ \highCut and \lowFlux set}{
246 \usedThis $\leftarrow$ false\;
247 \usedPrev $\leftarrow$ false\;
250 \total $\leftarrow$ \Current\;
251 \lIf{\lowCut $<$ \Previous $<$ \highCut and not \usedPrev}{
252 \total $\leftarrow$ \total + \Previous\;
254 \If{\lowCut $<$ \Next $<$ \highCut}{
255 \total $\leftarrow$ \total + \Next\;
256 \usedThis $\leftarrow$ true\;
259 \usedPrev $\leftarrow$ true\;
262 \usedPrev $\leftarrow$ false\;
263 \usedThis $\leftarrow$ false\;
267 Here, the function \FuncSty{GetHighCut} evaluates a Landau
268 distribution fitted to the energy spectrum in the $\eta$ bin
269 specified. It returns
273 where $\Delta_{mp}$ is the most probable energy loss, and $w$ is the
274 width of the Landau distribution.
276 The \KwSty{if} in line 5, says that if the previous strip was merged
277 with current one, and the signal of the current strip was added to
278 that, then we the current signal is set to 0, and we mark it as used
279 for the next iteration (\DataSty{previous signal
280 used}$\leftarrow$true).
282 The \KwSty{if} in line 10 checks if the current signal is smaller than
283 the next signal, the next signal is larger than the upper cut defined
284 above, and if we have a low--flux event. If that condition is met,
285 then the current signal is the smaller of two possible candidates for
286 merging, and it should be merged into the next signal. Note, that
287 this \emph{only} applies in low--flux events.
289 On line 15, we test if the previous signal lies between our low and
290 high cuts, and if it has not been marked as being used. If so, we add
291 it to our current signal.
293 The next \KwSty{if} on line 16 checks if the next signal is within our
294 cut bounds. If so, we add that signal to the current signal and mark
295 it as used for the next iteration (\DataSty{this signal
296 used}$\leftarrow$true). It will then be zero'ed on the next
297 iteration by the condition on line 6.
299 Finally, if our signal is still larger than 0, we return the signal
300 and mark this signal as used (\DataSty{previous signal
301 used}$\leftarrow$true) so that it will not be used in the next
302 iteration. Otherwise, we mark the current signal and the next signal
303 as unused and return a 0.
306 \subsection*{Density calculator}
308 The density calculator loops over all the strip signals and calculates
309 the inclusive (primary + secondary) charged particle density in
310 pre-defined $(\eta,\varphi)$ bins.
312 If the event is classified as a low--flux event, then the number of
313 charged particles in a given by a simple threshold:
317 0 & \Delta_t < \text{low cut}\\
318 1 & \Delta_t \ge \text{low cut}\\
319 \end{array}\right.\quad,
321 where $t$ is the strip identifier, $\Delta_t$ is the scaled energy
322 deposition in that strip, and 'low cut' is a predefined cut. For high
323 flux events, the number charged particles in a strip is calculated
324 using multiple Landau distributions fitted to the energy loss spectrum
325 at a given $\eta$ value.
327 \Delta_{2,mp} &= 2 \Delta_{mp}+ 2 w \log(2)\nonumber\\
328 \Delta_{3,mp} &= 3 \Delta_{mp}+ 3 w \log(2)\nonumber\\
329 N_{ch,t} &= \frac{\landau{\Delta_t,\Delta_{mp},w}+
330 2\,\alpha\,\landau{\Delta_t,\Delta_{2,mp},2w} +
331 3\,\beta\,\landau{\Delta_t,\Delta_{3,mp},3w}}{%
332 \landau{\Delta_t,\Delta_{mp},w}+
333 \alpha\,\landau{\Delta_t,\Delta_{2,mp},2w} +
334 \beta\,\landau{\Delta_t,\Delta_{3,mp},3w}}\quad,
336 where $\landau{x,\psi,W}$ is the evaluation of the Landau distribution
337 with most probable value $\psi$ and width $W$ at $x$, $w$ is the width
338 of the first MIP peak, $\Delta_{mp}$ the most probable value of
339 the first MIP peak, and $\alpha$ and
340 $\beta$ are the relative strength of the second and third MIP peak in
341 the fitted energy loss spectrum.
343 But before the signal $N_{ch,t}$ can be added to the $(\eta,\varphi)$
344 bin in one of the 5 per--ring histograms, it needs to be corrected for
345 the $\varphi$ acceptance of the strip, as well as a correction for
346 double hits in low--flux events.
348 The acceptance correction is only applicable where the strip length
349 does not cover the full sector. This is the case for the outer strips
350 in both the inner and outer type rings. The acceptance correction is
354 a_t &= \frac{l_t}{\Delta\varphi}\quad
356 where $l_t$ is the strip length in radians at constant $r$, and
357 $\Delta\varphi$ is $2\pi$ divided by the number of sectors in the
358 ring (20 for inner type rings, and 40 for outer type rings).
360 Even in low--flux events it is possible that more than one particle
361 hits a strip. However, for low--flux events, it is not possible to
362 reconstruct the 3\textsuperscript{rd} nor even the
363 2\textsuperscript{nd} MIP peak in the energy loss spectrum.
364 Therefore, the strip signal needs to be corrected to the average
365 number of particle impinging on a strip at a given $\eta$.
367 d_t &= \left\{\begin{array}{cl} \langle n_t\rangle & \text{low
373 The final $(\eta,\varphi)$ content of the 5 output vertex dependent,
374 per--ring histograms of the inclusive charged particle density is then
377 \dndetadphi[incl,r,v,i(\eta,\varphi)] &= \sum_t^{t\in(\eta,\varphi)}
380 where $t$ runs over the strips in the $(\eta,\varphi)$ bin.
382 \subsection*{Corrections}
384 The corrections code receives the five vertex dependent,
385 per--ring histograms of the inclusive charged particle density
386 $\dndetadphi[incl,r,v,i]$ from the density calculator and applies
389 \item[Secondary correction:] This is a 2 dimensional histogram
390 generated from simulations of the ratio of primary particles to the
391 total number of particles that fall within an $(\eta,\varphi)$ bin
394 \frac{N_{ch,\text{primary}}(\eta,\varphi)}{N_{ch}(\eta,\varphi)}\quad.
396 The $\eta$ and $\varphi$ of $N_{ch,primary}(\eta,\varphi)$ is given
397 by summing over the charged particles labelled as primaries \emph{at
398 the time of the collision} as defined in the simulation code.
399 That is, it is the number of primaries within the bin at the
400 collision point --- not at the \FMD{}. $N_{ch}(\eta,\varphi)$ is
401 evaluated as the sum of all charged particle that hit the FMD in the
402 given $(\eta,\varphi)$ bin.
403 \item[Event selection efficiency:]
404 \item[Sharing correction efficiency:]
407 The 5 output vertex dependent, per--ring histograms of the primary
408 charged particle density is then given by
410 \dndetadphi[r,v,i(\eta,\varphi)] &=
411 s(\eta,\varphi)\,e(\eta,\varphi)\,m(\eta)\dndetadphi[incl,r,v,i(\eta,\varphi)]
414 \subsection*{Histogram collector}
416 The histogram collector collects the information from the 5 vertex
417 dependent, per--ring histograms of the primary charged particle
418 density $\dndetadphi[r,v,i]$ into a single vertex dependent histogram
419 of the charged particle density $\dndetadphi[v,i]$.
421 To do this, it first calculates, for each vertex bin, the $\eta$ bin
422 range to use for each ring. It investigates the secondary correction
423 maps $s(\eta,\varphi)$ to find the edges of the map, and then applies
424 a safety margin of a few bins, to ensure that the data selected does
425 not have too large corrections associated with it.
427 It then loops over the bins in the defined $\eta$ range and sums the
428 contributions from each of the 5 histograms. If the $\eta$ ranges of
429 two rings overlap, then the collector calculates the average and adds
430 the errors in quadrature.
432 The output vertex dependent histogram of the primary
433 charged particle density is then given by
435 \dndetadphi[v,i(\eta,\varphi)] &=
436 \frac{1}{N_{r\in(\eta,\varphi)}}\sum_{r}^{r\in(\eta,\varphi)}
437 \dndetadphi[r,v,i(\eta,\varphi)]\\
438 \delta\left[\dndetadphi[v,i(\eta,\varphi)]\right] &=
439 \frac{1}{N_{r\in(\eta,\varphi)}}\sqrt{\sum_{r}^{r\in(\eta,\varphi)}
440 \delta\left[\dndetadphi[r,v,i(\eta,\varphi)]\right]^2}\\
443 where $N_{r\in(\eta,\varphi)}$ is the number of overlapping histograms
444 in the given $(\eta,\varphi)$ bin.
447 \section*{Building the final $\dndeta$}