Methods for flow analysis in ALICE FLOW package

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Methods for flow analysis in ALICE FLOW package
Ante BilandzicTrento, 15.09.2009
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Outline

• Anisotropic flow
• From theorists point of view
• From experimentalists point of view
• Multiparticle azimuthal correlations
• Methods for flow analysis implemented in ALICE
  flow package
• 2-particle methods
• Multiparticle methods (4-, 6- and 8- particle
  methods)
• Genuine multiparticle methods
• Recent development for ALICE Q-cumulants
• Method comparison
• Idealistic simulations on the fly
• Realistic pp simulations (Pythia)
• Realistic heavy-ion simulations (Therminator)

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Anisotropic flow (th)
Azimuthal distributions of particles
measured with respect to reaction plane (spanned
by impact parameter vector and beam axis) are not
isotropic.

• S. Voloshin and Y. Zhang (1996)

• Harmonics vn quantify anisotropic flow

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Anisotropic flow (exp)

• Since reaction plane cannot be measured e-b-e,
  consider the quantities which do not depend on
  its orientation multiparticle azimuthal
  correlations
• Basic underlying assumption of flow analysis If
  only flow correlations are present we can write

• Cool idea but already at this level there are two
  important issues
• Statistical flow fluctuations e-b-e, what we
  measure is actually
• Other sources of correlations (systematic bias
  a.k.a. nonflow)

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Methods to measure flow

• Measure the flow with rapidity gaps (using the
  PMD and FMDs)
• advantage most of nonflow is due to short range
  correlations, thus using rapidity gaps suppresses
  nonflow
• disadvantage not known how much nonflow is
  supressed, results are model dependent and "long
  range" rapidity correlations are not modeled very
  well
• Measure the deflection of the spectators at beam
  and target rapidity (v1 in the ZDC)
• advantages 1) nonflow is really very much
  suppressed, 2) fluctuations are also decoupled
  from midrapidity source
• disadvantage small resolution c, not an easy
  measurement
• Measure flow using multiparticle correlations

All three methods for measuring flow are used in
ALICE, but in remainder of the talk will focus
only on the last one
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Multiparticle azimuthal correlations

• Typically nonflow correlations involve only few
  particles. Based purely on combinatorial grounds
  

• One can use 2- and 4-particle correlations to
  estimate flow only if

• It is possible to obtain flow estimate from the
  genuine multiparticle correlation (Ollitrault et
  al). In this case one reaches the theoretical
  limit of applicability

• Can we now relax once we have devised
  multiparticle correlations to estimate flow
  experimentally?

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There are some more issues

• Basic problem How to calculate multiparticle
  correlations? Naïve approach leads to evaluation
  of nested loops over heavy-ion data, certainly
  not feasible
• Numerical stability of flow estimates?
• Measured azimuthal correlations are strongly
  affected by any inefficiencies in the detector
  acceptance
• Is one pass over data enough or not to correct
  for it?
• Can we also estimate subdominant flow harmonics?
• Besides the fact that flow fluctuates e-b-e, and
  very likely also the systematic bias coming from
  nonflow, the multiplicity fluctuates as well e-b-e

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In the rest of the talk

• Outline of the methods based on multiparticle
  azimuthal correlations which were developed by
  various authors to tackle all these issues and
  which were implemented in the ALICE FLOW package
• Emphasis will be given to cumulants (in
  particular to Q-cumulants a method recently
  developed for ALICE which is essentially just
  another way to calculate cumulants with potential
  improvements)
• Notation In what follows I will use frequently
  phrase non-weighted Q-vector evaluated in
  harmonic n for the following

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Methods implemented for ALICE(naming conventions)

• MCEP Monte Carlo Event Plane
• SP Scalar Product
• GFC Generating Function Cumulants
• QC Q-cumulants
• FQD Fitting q-distribution
• LYZ Lee-Yang Zero (sum and product)
• LYZEP Lee-Yang Zero Event Plane

• Raimond Snellings, Naomi van der Kolk, ab

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MCEP

• Using the knowledge of sampled reaction plane
  event-by-event and calculating directly

• Both integrated and differential flow calculated
  in this way
• Flow estimates of all other methods in
  simulations are being compared to this one

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Cumulants A principle

• Ollitrault et al Imagine that there are only
  flow and 2-particle nonflow correlations present.
  Than contributions to measured 2- and 4-particle
  correlations read

• By definition, for detectors with uniform
  acceptance 2nd and 4th order cumulant are given by

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Cumulants GFC

• To circumvent evaluation of nested loops to get
  multiparticle correlations Borghini, Dinh and
  Ollitrault proposed the usage of generating
  function used regularly at STAR (and recently
  at PHENIX)

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Cumulants GFC

• Example of numerical instability making
  equivalent simulations with fixed multiplicity M
  500 and statistics of N 105 events, but with
  different input values for flow

input v2 0.05
input v2 0.15
GFC method has 2 main limitations a) not
numerically stable for all values of
multiplicity, flow and number of events, b)
biased by flow fluctuations
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Cumulants QC

• Another approach to circumvent evaluation of
  nested loops to get multiparticle correlations
  Sergei Voloshins idea to express multiparticle
  correlations in terms of expressions involving
  Q-vectors evaluated (in general) in different
  harmonics
• Once you have expressed multiparticle
  correlations in this way, it is trivial to build
  up cumulants from them
• Publication S. Voloshin, R. Snellings, ab Flow
  analysis with Q-cumulants is in preparation

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Demystifying QC

• Define average 2- and 4-particle azimuthal
  correlations for a single event as

• Define average 2- and 4-particle azimuthal
  correlations for all events as

and follow the recipe
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QC recipe, part 1

• Evaluate Q-vector in harmonics n and 2n for a
  particular event and insert those quantities in
  the following Eqs

• These Eqs. give exactly the same answer for 2-
  and 4-particle correlations for a particular
  event as the one obtained with two and four
  nested loops, but in almost no CPU time

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QC recipe, part 2

• How to obtain exact averages for all events?
• By using multiplicity weights! For 2-particle
  correlation multiplicity weight is M(M-1) and for
  4-particle correlation multiplicity weight is
  M(M-1)(M-2)(M-3)

• Now it is trivial to build up 2nd and 4th order
  cumulant

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Method comparisons(series of plots)
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Nonflow

• Example input v2 0.05, M 500, N 5 106
  and simulate nonflow by taking each particle twice

As expected only 2-particle estimates are
biased
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Flow fluctuations

• If the flow fluctuations are Gaussian, the
  theorists say

• Example 1 v2 0.05 /- 0.02 (Gaussian), M
  500, N 106

Gaussian flow fluctuations affect the methods as
predicted
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Flow fluctuations

• Example 2 v2 in 0.04,0.06 (uniform), M 500,
  N 9 106

Uniform flow fluctuations affect the methods
differently as the Gaussian fluctuations
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Multiplicity fluctuations (small ltMgt)

• Example 1 M 50 /- 10 (Gaussian), input fixed
  v2 0.075, N 10 106

• LYZ (sum) big statistical spread, SP
  systematically biased
• FQD doing fine, spread for QC is smaller than for
  GFC

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Extracting subdominant harmonic

• Example input v1 0.10, v2 0.05, M 500, N
  10 106 and estimating subdominant harmonic v2

All methods are fine
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Extracting subdominant harmonic

• Example input v2 0.05, v4 0.10, M 500, N
  10 106 and estimating subdominant harmonic v2

FQD and LYZ (sum) are biased and we still have to
tune the LYZ product
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Non-uniform acceptance

• To correct for the bias on flow estimates coming
  from the non-uniform acceptance of the detector,
  several techniques were proposed by various
  authors flattening, recentering, etc.
• require additional run over data
• some of them not applicable for detectors with
  gaps in azimuthal acceptance (e.g. flattening)
• Ollitrault et al proposed evaluating generating
  functions along fixed directions in the
  laboratory frame and averaging the results
  obtained for those directions
• works fine for GFC and LYZ
• no need for an additional run over data
• Recent For Q-cumulants it is possible explicitly
  to calculate and subtract the bias coming from
  the non-uniform acceptance
• applicable to all types of non-uniform acceptance
• one run over data enough

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Non-uniform acceptance

• The terms in yellow counter balance the bias due
  to non-uniform acceptance, so that QC2 and
  QC4 remain unbiased

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Non-uniform acceptance

• Example input v2 0.05, M 500, N 8 106,
  particles emitted in 60o lt f lt 90o and 180o lt f lt
  225o ignored
• Detectors azimuthal acceptance has two gaps

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Non-uniform acceptance
Zoomed plot from LHS

• SP and FQD in its present form cannot be used if
  detector has gaps in acceptance
• QC6 and QC8 correction still not calculated
  and implemented, but the idea how to proceed is
  clear
• GFC and LYZ rely on averaging out the bias by
  making projections on 5 fixed directions
  pragmatic approach
• QC2 and QC4 the bias is explicitly
  calculated and subtracted

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Numerical stability

• Are estimates still numerically stable for very
  large flow?
• Example input v2 0.50, M 500, N 106

Zoomed plot from LHS

• LHS GFC estimates unstable (there is no unique
  set of points in a complex plain which give
  stable results for all values of number of
  events, average multiplicity and flow)
• RHS Methods not based on generating functions
  (SP and QC) are numerically much more stable

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QC factbook

• Possible to get both integrated and differential
  flow in a single run
• Not biased by interference between different
  harmonics can be applied to extract subdominant
  harmonics
• Not biased by interference between different
  order estimates for the same harmonic (e.g. you
  do not need the knowledge of the 8th order
  estimate to calculate the 2nd order estimate)
• Not biased by multiplicity fluctuations compared
  to GFC improved results for peripheral collisions
  
• Not biased by numerical errors compared to GFC
  no need to tune interpolating parameters (e.g. r0
  for GFC, QC has no parameters)
• Detector effects can be quantified and corrected
  for in a single run over data even for the
  detectors with gaps in azimuthal acceptance
• Biased by flow fluctuations

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Pythia pp

• Realistic pp data simulated with no flow
• ltMgt 10, N 3 104

• All multiparticle methods fail (because vn is not
  gtgt 1/M)
• ZDC will also fail for pp
• rapidity gaps do work albeit model dependent

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Therminator

• Realistic heavy-ion dataset (ltMgt 2164, N
  1728)

• Clear advantage of multiparticle methods over
  2-particle methods (GFC higher orders need tuning
  of interpolating parameters to suppress numerical
  instability)

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Therminator

• More detailed impression differential flow in pt

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Therminator

• More detailed impression differential flow in h

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Therminator

• Same dataset as before just reducing multiplicity
  with rapidity cuts to get to the more realistic
  values (ltMgt 634, N 1722)

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Heavy-ions in ALICE

• Assuming 100 minbias events/s during a run giving
  60k events in the first 10 minutes
• But a really safe estimate would be 10 ev/s on
  average during the whole PbPb run (2 weeks)

This shows that with a few minutes of good data
taking we can provide the first reliable
measurement of flow in ALICE
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Thanks!
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Backup slides
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FQD

• Evaluating event-by-event modulus of reduced flow
  vector and filling the histogram. The resulting
  distribution is fitted with the theoretical
  distribution in which flow appears as one of the
  parameters

• Method has 5 serious limitations a) cannot be
  used to obtain differential flow, b) theoretical
  distribution valid only for large multiplicities,
  c) cannot be used to extract the subdominant
  harmonic, d) cannot be used for detectors with
  gaps in azimuthal acceptance, e) biased by flow
  fluctuations

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FQD

• Example input v2 0.05, M 250, each particle
  taken twice to simulate 2-particle nonflow

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SP

• 2-particle method
• Using a magnitude of the flow vector as a weight

• un,i is the unit vector of the ith particle
  (which is excluded from the flow vector Qn)
• a and b denote flow vectors of two independent
  subevents

• Method has 4 serious limitations a) strongly
  biased by 2-particle nonflow correlations, b) in
  its present form biased by inefficiencies in
  detector acceptance, c) biased by multiplicity
  fluctuations, d) biased by flow fluctuations

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LYZ and LYZEP

• Introduced by Ollitrault et al
• Gives genuine multiparticle estimate, both for
  integrated and differential flow
• Two version implemented sum and product
• LYZEP additionally provides the event plane and
  it is based on LYZ (sum)
• The method has 3 main limitations a) one pass
  over data is not enough, b) not numerically
  stable for all flow values, c) biased by flow
  fluctuations

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LYZ product

• One should first compute for each event the
  complex-valued function

• Next one should average over events
  for each value of r and q

• For every q value one must then look for the
  position of the first positive minimum of
  the modulus

• This is the Lee-Yang zero and an estimate of the
  integrated flow is given now by

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LYZ sum

• Start by making the projection to an arbitrary
  laboratory angle q of the second-harmonic flow
  vector

• The sum generating function is given by

• The rest is analogous as in LYZ prod

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Demystifying QC

• How to use QC to calculate the differential flow?
  
• Denote angles of the particles belonging to the
  particular bin of interest with y and angles of
  particles used to determine the reaction plane
  with f
• Define average reduced 2- and 4-particle
  azimuthal correlations for a particular bin in a
  single event as

• Define average reduced 2- and 4-particle
  azimuthal correlations for a particular bin over
  all events as

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QC recipe, part 3

• Evaluate also Q-vector in harmonics n and 2n for
  particles belonging to the bin of interest in a
  single event and denote it is as qn and q2n. Plug
  Qn , Q2n , qn and q2n into

• M is the multiplicity of event and m is the
  multiplicity of
• particles in a particular bin in that event

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QC recipe, part 4

• To get the final average for reduced 2- and
  4-particle correlations over all events use the
  slightly modified multiplicity weights

• These Eqs. give exactly the same answer for
  reduced 2- and 4-particle correlations over all
  events as the one obtained with two and four
  nested loops, but in almost no CPU time

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QC recipe, the final touch

• Build up the cumulants for differential flow in
  the spirit of Ollitrault et al

• and estimate differential flow from them