Methods for flow analysis in Methods for flow analysis in ALICE FLOW packageALICE FLOW package

Ante BilandzicAnte Bilandzic

Trento,Trento, 15.09.200915.09.2009

22OutlineOutline Anisotropic flowAnisotropic flow

From theorists’ point of viewFrom theorists’ point of view From experimentalists’ point of viewFrom experimentalists’ point of view

Multiparticle azimuthal correlations Multiparticle azimuthal correlations Methods for flow analysis implemented in ALICE flow Methods for flow analysis implemented in ALICE flow

packagepackage 2-particle methods2-particle methods Multiparticle methods (4-, 6- and 8- particle methods)Multiparticle methods (4-, 6- and 8- particle methods) Genuine multiparticle methods Genuine multiparticle methods Recent development for ALICE: Recent development for ALICE: QQ-cumulants-cumulants

Method comparisonMethod comparison Idealistic simulations ‘on the fly’ Idealistic simulations ‘on the fly’ Realistic Realistic pppp simulations (Pythia) simulations (Pythia) Realistic heavy-ion simulations (Therminator)Realistic heavy-ion simulations (Therminator)

33Anisotropic flow (th)Anisotropic flow (th)

x

yz

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yz

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yz

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S. Voloshin and Y. Zhang (1996):S. Voloshin and Y. Zhang (1996):

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

Harmonics Harmonics vvnn quantify anisotropic flow quantify anisotropic flow

44Anisotropic flow (exp)Anisotropic flow (exp) Since reaction plane cannot be measured e-b-e, consider Since reaction plane cannot be measured e-b-e, consider

the quantities which do not depend on it’s orientation: the quantities which do not depend on it’s orientation: multiparticle azimuthal correlationsmultiparticle azimuthal correlations

Basic underlying assumption of flow analysis: If only flow Basic underlying assumption of flow analysis: If only flow correlations are present we can writecorrelations are present we can write

Cool idea but already at this level there are two important Cool idea but already at this level there are two important issuesissues Statistical flow fluctuations e-b-e, what we measure is actually:Statistical flow fluctuations e-b-e, what we measure is actually:

Other sources of correlations (systematic bias a.k.a. nonflow):Other sources of correlations (systematic bias a.k.a. nonflow):

55Methods to measure flowMethods to measure flow1)1) Measure the flow with rapidity gaps (using the PMD and Measure the flow with rapidity gaps (using the PMD and

FMDs) FMDs) advantage: most of nonflow is due to short range correlations, advantage: most of nonflow is due to short range correlations,

thus using rapidity gaps suppresses nonflowthus using rapidity gaps suppresses nonflow disadvantage: not known how much nonflow is supressed, disadvantage: not known how much nonflow is supressed,

results are model dependent and "long range" rapidity results are model dependent and "long range" rapidity correlations are not modeled very well correlations are not modeled very well

2)2) Measure the deflection of the spectators at beam and Measure the deflection of the spectators at beam and target rapidity (target rapidity (vv11 in the ZDC) in the ZDC)

advantages: 1) nonflow is really very much suppressed, 2) advantages: 1) nonflow is really very much suppressed, 2) fluctuations are also decoupled from midrapidity source fluctuations are also decoupled from midrapidity source

disadvantage: small resolution disadvantage: small resolution , not an easy measurement , not an easy measurement

3)3) Measure flow using multiparticle correlations 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

66Multiparticle azimuthal correlationsMultiparticle azimuthal correlations Typically nonflow correlations involve only few particles. Typically nonflow correlations involve only few particles.

Based purely on combinatorial grounds: Based purely on combinatorial grounds:

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

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

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

77There are some more issues…There are some more issues… Basic problem: How to calculate multiparticle Basic problem: How to calculate multiparticle

correlations? Naïve approach leads to evaluation of correlations? Naïve approach leads to evaluation of nested loops over heavy-ion data, certainly not feasible nested loops over heavy-ion data, certainly not feasible Numerical stability of flow estimates?Numerical stability of flow estimates?

Measured azimuthal correlations are strongly affected by Measured azimuthal correlations are strongly affected by any inefficiencies in the detector acceptanceany inefficiencies in the detector acceptance Is one pass over data enough or not to correct for it?Is one pass over data enough or not to correct for it?

Can we also estimate subdominant flow harmonics?Can we also estimate subdominant flow harmonics? Besides the fact that flow fluctuates e-b-e, and very likely Besides the fact that flow fluctuates e-b-e, and very likely

also the systematic bias coming from nonflow, the also the systematic bias coming from nonflow, the multiplicity fluctuates as well e-b-emultiplicity fluctuates as well e-b-e

88In the rest of the talkIn the rest of the talk Outline of the methods based on multiparticle azimuthal Outline of the methods based on multiparticle azimuthal

correlations which were developed by various authors to correlations which were developed by various authors to tackle all these issues and which were implemented in tackle all these issues and which were implemented in the ALICE FLOW packagethe ALICE FLOW package Emphasis will be given to cumulants (in particular to Emphasis will be given to cumulants (in particular to QQ--

cumulants – a method recently developed for ALICE which is cumulants – a method recently developed for ALICE which is essentially just another way to calculate cumulants with potential essentially just another way to calculate cumulants with potential improvements)improvements)

Notation: In what follows I will use frequently phrase Notation: In what follows I will use frequently phrase “non-weighted “non-weighted QQ-vector-vector evaluated in harmonic evaluated in harmonic nn” ” for the for the following:following:

99

Methods implemented for ALICEMethods implemented for ALICE(naming conventions)(naming conventions)

MCEP = Monte Carlo Event PlaneMCEP = Monte Carlo Event Plane SP = Scalar ProductSP = Scalar Product GFC = Generating Function CumulantsGFC = Generating Function Cumulants QC = QC = QQ-cumulants -cumulants FQD = Fitting FQD = Fitting qq-distribution-distribution LYZ = Lee-Yang Zero (sum and product)LYZ = Lee-Yang Zero (sum and product) LYZEP = Lee-Yang Zero Event PlaneLYZEP = Lee-Yang Zero Event Plane

Raimond Snellings, Naomi van der Kolk, abRaimond Snellings, Naomi van der Kolk, ab

1010MCEPMCEP Using the knowledge of sampled reaction plane event-Using the knowledge of sampled reaction plane event-

by-event and calculating directlyby-event and calculating directly

Both integrated and differential flow calculated in this Both integrated and differential flow calculated in this wayway

Flow estimates of all other methods in simulations are Flow estimates of all other methods in simulations are being compared to this one being compared to this one

1111Cumulants: A principleCumulants: A principle Ollitrault Ollitrault et alet al: Imagine that there are only flow and 2-: Imagine that there are only flow and 2-

particle nonflow correlations present. Than contributions particle nonflow correlations present. Than contributions to measured 2- and 4-particle correlations readto measured 2- and 4-particle correlations read

By definition, for detectors with uniform acceptance 2By definition, for detectors with uniform acceptance 2ndnd and and 44thth order cumulant are given by order cumulant are given by

1212Cumulants: GFCCumulants: GFC To circumvent evaluation of nested loops to get To circumvent evaluation of nested loops to get

multiparticle correlations: Borghini, Dinh and Ollitrault multiparticle correlations: Borghini, Dinh and Ollitrault proposed the usage of generating function – used proposed the usage of generating function – used regularly at STAR (and recently at PHENIX):regularly at STAR (and recently at PHENIX):

1313Cumulants: GFCCumulants: GFC Example of numerical instability: making equivalent Example of numerical instability: making equivalent

simulations with fixed multiplicity simulations with fixed multiplicity M = M = 500500 and statistics and statistics of of NN = 10 = 1055 events, but with different input values for flow events, but with different input values for flow

input input vv22 = 0.05 = 0.05 input: input: vv22 = 0.15 = 0.15

GFC method has 2 main limitations: a) not numerically stable for allGFC method has 2 main limitations: a) not numerically stable for allvalues of multiplicity, flow and number of events, b) biased by flowvalues of multiplicity, flow and number of events, b) biased by flowfluctuationsfluctuations

1414Cumulants: QCCumulants: QC Another approach to circumvent evaluation of nested Another approach to circumvent evaluation of nested

loops to get multiparticle correlations: Sergei Voloshin’s loops to get multiparticle correlations: Sergei Voloshin’s idea to express multiparticle correlations in terms of idea to express multiparticle correlations in terms of expressions involving expressions involving QQ-vectors evaluated (in general) in -vectors evaluated (in general) in different harmonicsdifferent harmonics

Once you have expressed multiparticle correlations in Once you have expressed multiparticle correlations in this way, it is trivial to build up cumulants from themthis way, it is trivial to build up cumulants from them

Publication S. Voloshin, R. Snellings, ab Publication S. Voloshin, R. Snellings, ab “Flow analysis “Flow analysis with with QQ-cumulants”-cumulants” is in preparation is in preparation

1515Demystifying QCDemystifying QC Define average 2- and 4-particle azimuthal correlations Define average 2- and 4-particle azimuthal correlations

for a single event asfor a single event as

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

and follow the recipe…and follow the recipe…

1616

Evaluate Evaluate QQ--vector in harmonics vector in harmonics nn and and 2n2n for a particular for a particular event and insert those quantities in the following Eqs:event and insert those quantities in the following Eqs:

QC recipe, part 1QC recipe, part 1

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

1717

How to obtain How to obtain exactexact averages for all events? averages for all events? By using multiplicity weights! For 2-particle correlation By using multiplicity weights! For 2-particle correlation

multiplicity weight is multiplicity weight is M(M-1)M(M-1) and for 4-particle correlation and for 4-particle correlation multiplicity weight is multiplicity weight is M(M-1)(M-2)(M-3)M(M-1)(M-2)(M-3)

QC recipe, part 2QC recipe, part 2

Now it is trivial to build up 2Now it is trivial to build up 2ndnd and 4 and 4thth order cumulant order cumulant

1818

Method comparisonsMethod comparisons(series of plots)(series of plots)

1919NonflowNonflow

As expected only 2-particle estimates are biasedonly 2-particle estimates are biased

Example: input Example: input vv22 = 0.05, = 0.05, MM = 500 = 500, , NN = 5 = 5 ×× 10 1066 and and

simulate nonflow by taking each particle twicesimulate nonflow by taking each particle twice

2020Flow fluctuationsFlow fluctuations

Example 1: Example 1: vv22 = = 0.05 +/- 0.020.05 +/- 0.02 (Gaussian), (Gaussian), MM = 500, = 500, NN = 10 = 1066

Gaussian flow fluctuations affect the methods as predictedGaussian flow fluctuations affect the methods as predicted

If the flow fluctuations are Gaussian, the theorists sayIf the flow fluctuations are Gaussian, the theorists say

2121Flow fluctuationsFlow fluctuations Example 2: Example 2: vv22 in in [0.04,0.06][0.04,0.06] (uniform), (uniform), MM = 500, = 500, NN = 9 = 9 ×× 10 1066

Uniform flow fluctuations affect the methods differently as the Gaussian fluctuationsUniform flow fluctuations affect the methods differently as the Gaussian fluctuations

2222Multiplicity fluctuations (small <M>)Multiplicity fluctuations (small <M>) Example 1: Example 1: MM = 50 +/- 10 = 50 +/- 10 (Gaussian), input fixed (Gaussian), input fixed vv22 = =

0.075, 0.075, NN = 10 = 10 ×× 10 1066

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

2323Extracting subdominant harmonicExtracting subdominant harmonic Example: input Example: input vv11 = 0.10, = 0.10, vv22 = 0.05, = 0.05, MM = 500 = 500, , NN = 10 = 10 ×× 10 1066

and estimating subdominant harmonic and estimating subdominant harmonic vv22

All methods are fine

2424Extracting subdominant harmonicExtracting subdominant harmonic Example: input Example: input vv22 = 0.05, = 0.05, vv44 = 0.10, = 0.10, MM = 500, = 500, NN = 10 = 10 ×× 10 1066

and estimating subdominant harmonic and estimating subdominant harmonic vv22

FQD and LYZ (sum) are biased and we still have to tune FQD and LYZ (sum) are biased and we still have to tune the LYZ productthe LYZ product

2525Non-uniform acceptanceNon-uniform acceptance To correct for the bias on flow estimates coming from the To correct for the bias on flow estimates coming from the

non-uniform acceptance of the detector, several non-uniform acceptance of the detector, several techniques were proposed by various authors: flattening, techniques were proposed by various authors: flattening, recentering, etc. recentering, etc. require additional run over datarequire additional run over data some of them not applicable for detectors with gaps in azimuthal some of them not applicable for detectors with gaps in azimuthal

acceptance (e.g. flattening)acceptance (e.g. flattening) Ollitrault Ollitrault et al et al proposed evaluating generating functions proposed evaluating generating functions

along fixed directions in the laboratory frame and along fixed directions in the laboratory frame and averaging the results obtained for those directions:averaging the results obtained for those directions: works fine for GFC and LYZ works fine for GFC and LYZ no need for an additional run over datano need for an additional run over data

Recent: For Recent: For QQ-cumulants it is possible explicitly to -cumulants it is possible explicitly to calculate and subtract the bias coming from the non-calculate and subtract the bias coming from the non-uniform acceptanceuniform acceptance applicable to all types of non-uniform acceptanceapplicable to all types of non-uniform acceptance one run over data enough one run over data enough

2626Non-uniform acceptanceNon-uniform acceptance

The terms in The terms in yellow counter balance the bias due to non-uniform counter balance the bias due to non-uniform acceptance, so that acceptance, so that QCQC{2}{2} and and QCQC{4}{4} remain unbiased remain unbiased

2727Non-uniform acceptanceNon-uniform acceptance Example: input Example: input vv22 = 0.05, = 0.05, MM = 500, = 500, NN = 8 = 8 ×× 10 1066, particles , particles

emitted inemitted in 60 60oo << < 90< 90oo and and 180180oo < < < 225< 225oo ignored ignored Detector’s azimuthal acceptance has two gaps:Detector’s azimuthal acceptance has two gaps:

2828Non-uniform acceptanceNon-uniform acceptance

SP and FQD in its present form cannot be used if detector has gaps in SP and FQD in its present form cannot be used if detector has gaps in acceptanceacceptance

QCQC{6}{6} and and QCQC{8}:{8}: correction still not calculated and implemented, but the correction still not calculated and implemented, but the idea how to proceed is clear idea how to proceed is clear

GFC and LYZ rely on averaging out the bias by making projections on 5 GFC and LYZ rely on averaging out the bias by making projections on 5 fixed directions – pragmatic approach fixed directions – pragmatic approach

QCQC{2}{2} and and QCQC{4}:{4}: the bias is explicitly calculated and subtracted the bias is explicitly calculated and subtracted

Zoomed plot from LHS: Zoomed plot from LHS:

2929Numerical stabilityNumerical stability Are estimates still numerically stable for very large flow? Are estimates still numerically stable for very large flow? Example: input Example: input vv22 = 0.50, = 0.50, MM = 500, = 500, NN = 10 = 1066

LHS: GFC estimates unstable (there is no unique set of points in a 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 complex plain which give stable results for all values of number of events, average multiplicity and flow)events, average multiplicity and flow)

RHS: Methods not based on generating functions (SP and QC) are RHS: Methods not based on generating functions (SP and QC) are numerically much more stable numerically much more stable

Zoomed plot from LHS: Zoomed plot from LHS:

3030QC factbookQC factbook Possible to get both integrated and differential flow in a Possible to get both integrated and differential flow in a

single run single run Not biased by interference between different harmonics: Not biased by interference between different harmonics:

can be applied to extract subdominant harmonics can be applied to extract subdominant harmonics Not biased by interference between different order Not biased by interference between different order

estimates for the same harmonic (e.g. you do not need estimates for the same harmonic (e.g. you do not need the knowledge of the 8the knowledge of the 8thth order estimate to calculate the order estimate to calculate the 22ndnd order estimate) order estimate)

Not biased by multiplicity fluctuations: compared to GFC Not biased by multiplicity fluctuations: compared to GFC improved results for peripheral collisions improved results for peripheral collisions

Not biased by numerical errors: compared to GFC no Not biased by numerical errors: compared to GFC no need to tune interpolating parameters (e.g. need to tune interpolating parameters (e.g. rr00 for GFC, for GFC, QC has no parameters)QC has no parameters)

Detector effects can be quantified and corrected for in a Detector effects can be quantified and corrected for in a single run over data even for the detectors with gaps in single run over data even for the detectors with gaps in azimuthal acceptanceazimuthal acceptance

Biased by flow fluctuations

3131Pythia ppPythia pp Realistic pp data simulated with no flowRealistic pp data simulated with no flow <M><M> ~ 10~ 10, , NN = 3 = 3 ×× 10 1044

All multiparticle methods fail (because All multiparticle methods fail (because vvnn is not is not >> 1/M>> 1/M))

ZDC will also fail for ppZDC will also fail for pp rapidity gaps do work albeit model dependent rapidity gaps do work albeit model dependent

3232TherminatorTherminator Realistic heavy-ion dataset (Realistic heavy-ion dataset (<M> = 2164, <M> = 2164, NN = 1728 = 1728):):

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

3333TherminatorTherminator More detailed impression: differential flow in More detailed impression: differential flow in pptt

3434TherminatorTherminator More detailed impression: differential flow in More detailed impression: differential flow in

3535TherminatorTherminator Same dataset as before just reducing multiplicity with Same dataset as before just reducing multiplicity with

rapidity cuts to get to the more realistic values rapidity cuts to get to the more realistic values (<M> = 634, (<M> = 634, NN = 1722 = 1722):):

3636Heavy-ions in ALICEHeavy-ions in ALICE

Assuming 100 minbias events/s Assuming 100 minbias events/s during a run giving 60k events in during a run giving 60k events in the first 10 minutes the first 10 minutes

But a really safe estimate would But a really safe estimate would be 10 ev/s on average during be 10 ev/s on average during the whole PbPb run (2 weeks) 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

3737

Thanks!Thanks!

3838

Backup slidesBackup slides

3939FQDFQD Evaluating event-by-event modulus of reduced flow Evaluating event-by-event modulus of reduced flow

vector and filling the histogram. The resulting distribution vector and filling the histogram. The resulting distribution is fitted with the theoretical distribution in which flow is fitted with the theoretical distribution in which flow appears as one of the parametersappears as one of the parameters

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

4040FQDFQD Example: input Example: input vv22 = 0.05, = 0.05, MM = 250, each particle taken = 250, each particle taken

twice to simulate 2-particle nonflow:twice to simulate 2-particle nonflow:

4141SPSP

uun,in,i is the unit vector of the is the unit vector of the iithth particle (which is excluded particle (which is excluded from the flow vector from the flow vector QQnn))

aa and and bb denote flow vectors of two independent subeventsdenote flow vectors of two independent subevents

2-particle method2-particle method Using a magnitude of the flow vector as a weight:Using a magnitude of the flow vector as a weight:

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

4242LYZ and LYZEPLYZ and LYZEP Introduced by Ollitrault Introduced by Ollitrault et alet al Gives genuine multiparticle estimate, both for integrated Gives genuine multiparticle estimate, both for integrated

and differential flowand differential flow Two version implemented – sum and productTwo version implemented – sum and product LYZEP additionally provides the event plane and it is LYZEP additionally provides the event plane and it is

based on LYZ (sum)based on LYZ (sum) The method has 3 main limitations: a) one pass over The method has 3 main limitations: a) one pass over

data is not enough, b) not numerically stable for all flow data is not enough, b) not numerically stable for all flow values, c) biased by flow fluctuationsvalues, c) biased by flow fluctuations

4343LYZ productLYZ product One should first compute for each event the complex-One should first compute for each event the complex-

valued function:valued function:

Next one should average Next one should average over events for each over events for each value of value of rr and and ::

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

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

4444LYZ sumLYZ sum Start by making the projection to an arbitrary laboratory Start by making the projection to an arbitrary laboratory

angle angle of the second-harmonic flow vector of the second-harmonic flow vector

The sum generating function is given byThe sum generating function is given by

The rest is analogous as in LYZ prodThe rest is analogous as in LYZ prod

4545Demystifying QCDemystifying QC How to use QC to calculate the differential flow? How to use QC to calculate the differential flow? Denote angles of the particles belonging to the particular Denote angles of the particles belonging to the particular

bin of interest with bin of interest with and angles of particles used to and angles of particles used to determine the reaction plane with determine the reaction plane with

Define average reduced 2’- and 4’-particle azimuthal Define average reduced 2’- and 4’-particle azimuthal correlations for a particular bin in a single event ascorrelations for a particular bin in a single event as

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

4646QC recipe, part 3QC recipe, part 3 Evaluate also Evaluate also Q-Q-vector in harmonics vector in harmonics nn and and 2n2n for for

particles belonging to the bin of interest in a single event particles belonging to the bin of interest in a single event and denote it is as and denote it is as qqnn and and qq2n2n. Plug . Plug QQn n ,, QQ2n 2n ,, qqn n andand qq2n2n intointo

MM is the multiplicity of event and is the multiplicity of event and mm is the multiplicity of is the multiplicity of particles in a particular bin in that eventparticles in a particular bin in that event

4747QC recipe, part 4QC recipe, part 4 To get the final average for reduced 2’- and 4’-particle To get the final average for reduced 2’- and 4’-particle

correlations over all events use the slightly modified correlations over all events use the slightly modified multiplicity weights:multiplicity weights:

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

4848QC recipe, the final touchQC recipe, the final touch

and estimate differential flow from them:and estimate differential flow from them:

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