critical point-onset april 2005m. j. tannenbaum 1/55 event-by-event fluctuations in heavy ion...

Critical Point-Onset April 2005 M. J. Tannenbaum 1/55

Event-by-Event Fluctuations Event-by-Event Fluctuations in Heavy Ion Collisionsin Heavy Ion Collisions

M. J. TannenbaumBrookhaven National Laboratory

Upton, NY 11973 USA2nd International Workshop on the Critical

Point and Onset of Deconfinement Bergen, Norway

April 1, 2005


or or I don’t know much about Statistical MechanicsI don’t know much about Statistical Mechanics

but I’m really good at Statistics!but I’m really good at Statistics!

M. J. TannenbaumBrookhaven National Laboratory

Upton, NY 11973 USA2nd International Workshop on the Critical

Point and Onset of Deconfinement Bergen, Norway

April 1, 2005


A Quick Course in StatisticsA Quick Course in Statistics• A StatisticStatistic is a quantity computed from a samplesample (which is drawn at randomrandom from a populationpopulation). A statistic is any function of the observed sample values.

• In physics we also call a population a probability density function, typically f(x)

• Two of the most popular statistics are the sum and the average

• Another popular statistic is the sample variance

where xi are the results of n repeated independent trials from the same population.


A Quick Course in Probability - IA Quick Course in Probability - I

• It is important to distinguish probability--which refers to properties or functions of the population, from statistics--which refers to properties or functions of the sample, although this distinction is often blurred (but not by statisticians).

• The probability density functions f(x) must be normalized so that the total probability for all possible outcomes is 1.

• The most popular probability computation is the expectation value or the mean:

• note that average, , is a property of the sample mean, , is a property of the population


Probability--IIProbability--II

• The mean or expectation value of a Statistic is often discussed:

• Of note is the biased expectation value of the sample variance:

where is the variance of the population

and the mean of the population is

and the variance of the average is

note the difference

is the standard deviation


Probability-III-sumsProbability-III-sumsconvolutionsconvolutions

• From the theory of mathematical statistics, the probability distribution of a random variable S(n) which is itself the sum of n independent random variables with a common distribution function f(x):

is given by fn(x), the n-fold convolution of the distribution f(x):

The mean, n=<S(n)> and standard deviation, n , of the n-fold convolution obey the familiar rule

where and are the mean and standard deviation of the distribution f(x).


Example-Example-EETT distributions distributions

• ET is an event-by-event variable which is a sum (S(n))

• The sum is over all particles emitted on an event into a fixed but large solid angle (which is different in every experiment)

• Measured in hadronic and electromagnetic calorimeters and even as the sum of charged particles i |pTi|

• Uses Gamma distribution as the pdf for ET on 1 collision=2 participants

• If ET adds independently for n collisions, participants, etc, the pdf is the n-fold convolution of f(x): pnp bb


NA5 (CERN) (1980) First ENA5 (CERN) (1980) First ETT dist. pp dist. pp

NA5 300 GeV PLB 112, 173 (1980)2, -0.88<y<0.67 NO JETS! s=23.7 GeVFit (by me) is dist p= 2.39 ± 0.06

UA1 (1982) (C.Rubbia) s=540 GeV. No Jets because ET is like multiplicity (n), composed of many soft particles near <pT> ! CERN-EP-82/122.

OOPS UA2 discovers jets 5 orders of magnitude down ET distribution!


First RHI data NA35 (NA5 Calorimeter) First RHI data NA35 (NA5 Calorimeter) CERN CERN 1616O+Pb O+Pb ssNNNN=19.4 GeVmidrapidity=19.4 GeVmidrapidity

p+Au is a dist p=3.36 Upper Edge of O+Pb is 16 convolutions of p+Au. WPNM!!PLB 184, 271 (1987)

WPN=Wounded Projectile Nucleon=projectile participant


PLB 197, 285 (1987) ZPC 38, 35 (1988)

E802-O+Au, O+CuE802-O+Au, O+Cumidrapidity at AGS midrapidity at AGS

ssNNNN=5.4GeV=5.4GeV

WPNM works in detailWPNM works in detail

• Maximum energy in O+Cu ~ same as O+Au--Upper edge of O+Au identical to O+Cu d/dE * 6

• Indicates large stopping at AGS 16O projectiles stopped in Cu so that energy emission (mid-rapidity) ceases

• Full O+Cu and O+Au spectra described in detail by WPNM based on measured p+Au


E802-AGSE802-AGSMidrapidity stopping!Midrapidity stopping!pBe & pAu have same pBe & pAu have same shape at midrapidity shape at midrapidity

over a wide range of over a wide range of PRC 63, 064602 (2001)

• confirms previous measurement PRC 45, 2933 (1992) that pion distribution from second collision shifts by > 0.8 units in y, out of aperture. Explains WPNM.


Collision Centrality MeasurementCollision Centrality MeasurementZeroDegreeCalorimeterZeroDegreeCalorimeter

0-5%

5-10%

10-15%

spectatorsparticipants

PHENIX at RHIC Au+Au-ZDC is biased

WA80 O+Au CERN


Extreme-Independent Extreme-Independent or Wounded Nucleon Modelsor Wounded Nucleon Models

• Number of Spectators (i.e. non-participants) Ns can be measured directly in Zero Degree Calorimeters (more complicated in Colliders)

• Enables unambiguous measurement of (projectile) participants = Ap -Ns

• For symmetric A+A collision Npart=2 Nprojpart

• Uncertainty principle and time dilation prevent cascading of produced particles in relativistic collisions h/mπc > 10fm even at AGS energies: particle production takes place outside the Nucleus in a p+A reaction.

• Thus, Extreme-Independent models separate the nuclear geometry from the dynamics of particle production. The Nuclear Geometry is represented as the relative probability per B+A interaction wn for a given number of total participants (WNM), projectile participants (WPNM), wounded projectile quarks (AQM), or other fundamental element of particle production.

• The dynamics of the elementary underlying process is taken from the data: e.g. the measured ET distribution for a p-p collision represents, 2 participants, 1 n-n collision, 1 wounded projectile nucleon, a predictable convolution of quark-nucleon collisions.


WA80 proof of Wounded Nucleon WA80 proof of Wounded Nucleon Model at 60, 200 A GeV using ZDCModel at 60, 200 A GeV using ZDC

= <Npart>

RA= <n>pA/ <n>pp= (1+<v>) / 2

<Npart>pA

<Npart>pp

Original Discovery by W. Busza, et alat FNAL <n>pA vs <> (Ncoll) PRD 22, 13 (1980)

PRC 44, 2736 (1991)


ISR-BCMOR-pp,dd,ISR-BCMOR-pp,dd, ssNNNN=31GeV WNM FAILS!=31GeV WNM FAILS!

WNM, AQM T.Ochiai, ZPC35,209(86)

PLB168, 158 (86)

Note WNM edge is parallel to p-p data!


But-Gamma Dist. fits uncover But-Gamma Dist. fits uncover Scaling in the mean over10 decades??Scaling in the mean over10 decades??

p-p p=2.50±0.06 - p=2.48±0.05 Is it Physics or a Fluke?


Summary of Wounded Nucleon ModelsSummary of Wounded Nucleon Models

• The classical Wounded Nucleon (Npart) Model (WNM) of Bialas, Bleszynski and Czyz (NPB 111, 461 (1976) ) works only at CERN fixed target energies, sNN~20 GeV.

• WNM overpredicts at AGS energies sNN~ 5 GeV (WPNM works at mid-rapidity)--this is due to stopping, second collision gives only few particles which are far from mid-rapidity. E802

• WNM underpredicts for sNN ≥ 31 GeV---is it Additive Quark Model? BCMOR

• This is the explanation of the ‘famous’ kink, well known as p+A effect since QM87+QM84


i.e. The kink is a p+A effect i.e. The kink is a p+A effect well known since 1987-seen at FNAL,ISR,AGSwell known since 1987-seen at FNAL,ISR,AGS


EETT systematics beyond the “kink” systematics beyond the “kink”

• In generic terms, dET/d implies a measurement corrected for:

Hadronic response---correct to E-mN for baryons, E+mN for antibaryons and E for all other hadrons.

ET corrected to =2, =1.0, scaling linearly in x

• For fixed target dET/dy=dET/d

• For collider at mid-rapidity dET/dy=1.2 x dET/d

• “Central collisions” varies from 2.5%-ile to 0.5%-ile in different experiments--try to correct to average 0-5%-ile (PHENIX definition)


NA35-->NA49 Pb+Pb NA35-->NA49 Pb+Pb ssNNNN=17 GeV=17 GeV

ET(2.1-3.4)--> dET/d=405 GeV@sNN=17 GeV

PRL 75, 3814 (1995)


PHENIX and E802 EPHENIX and E802 ETT compared compared

E877 dET/d=200 GeV@sNN=4.8 GeV PHENIX dET/d~606 GeV@sNN=200 GeV

PHENIX preliminary

E802 dET/d=128 GeV

= 22.5o = 2 x 22.5o

= 3 x 22.5o = 4 x 22.5o = 5 x 22.5o


Au+Au EAu+Au ETT spectra at AGS and RHIC are the same shape!!! spectra at AGS and RHIC are the same shape!!!


dEdETT/dy vs /dy vs ssNNNN for “central collisions” for “central collisions”

• Lines are pp s dependence. Lots of systematic issues but still kinky.

Bj

GeV

/fm

3

Note that Bj at sNN=20 GeV is the same in O+Au and Pb+Pb


EETT has a dimension. has a dimension.

Let’s now consider Let’s now consider number distributions number distributions

which are more typical of which are more typical of statisticsstatistics


What you have to rememberWhat you have to remember

• The mean and standard deviation of an average of n independent trials from the same population obey the rules:

where is the mean and x (or ) is the standard deviation of the population x .


Moments instead of distributionsMoments instead of distributions

• Sometimes I will discuss the probability distribution functions in detail, e.g. Binomial, Negative Binomial, Gamma Distribution

• More often I, as well as most others, will just use the first two moments, the mean and standard deviation (or variance=std2)

• It will become important to use combinations of moments which vanish for the case of zero correlation. The second “normalized binomial cumulant” or

vanishes for a poisson distribution, with no correlations.

• Most people use the normalized variance which is 1 for a poisson. It has its purpose, but not what everybody thinks.


Charged particle number fluctuationsCharged particle number fluctuations

“Particle number fluctuations in a canonical ensemble” V.V. Begun et al, PRC70, 034901 (2004)

NA49-BariConf-JPConf 5 (2005) 74

NB

DB

inom

ialPoisson

-

+

All


Binomial DistributionBinomial Distribution

• A Binomial distribution is the result of repeated independent trials, each with the same two possible outcomes: success, with probability p, and failure, with probability q=1-p. The probability for m successes on n trials (m,n 0) is:

The moments are:

• Example: distributing a total number of particles N onto a limited acceptance. Note that if p 0 with =np=constant we get a


Poisson DistributionPoisson Distribution• A Poisson distribution is the limit of the Binomial Distribution for a large number of independent trials, n, with small probability of success p such that the expectation value of the number of successes =<m>=np remains constant, i.e. the probability of m counts when you expect .

Moments:

• Example: The Poisson Distribution is intimately linked to the exponential law of Radioactive Decay of Nuclei, the time distribution of nuclear disintegration counts, giving rise to the common usage of the term “statistical fluctuations” to describe the Poisson statistics of such counts. The only assumptions are that the decay probability/time of a nucleus is constant, is the same for all nuclei and is independent of the decay of other nuclei.


Negative Binomial DistributionNegative Binomial Distribution

• For statisticians, the Negative Binomial Distribution represents the first departure from statistical independence of rare events, i.e. the presence of correlations. There is a second parameter 1/k, which represents the correlation: NBD Poisson as k , 1/k0

Moments:

The n-th convolution of NBD is an NBD with k nk, n such that /k remains constant. Hence constant 2/ vs Npart means multiplicity added by each participant is independent.

• Example: Multiplicity Distributions in p+p are Negative Binomial


UA5--Multiplicity Distributions in (small) UA5--Multiplicity Distributions in (small) intervals |intervals ||<|<cc around mid-rapidity are NBD around mid-rapidity are NBD

UA5 PLB 160, 193,199 (1985); 167, 476 (1986) Distributions are Negative Binomial, NOT POISSON: implies correlations

s=540 GeV


k vs k vs =2=2c c andand ss

• Distributions are never poisson at any s and s and • Something fishy with NA49 p+p result


NBD in O+Cu central collisions at AGS vs NBD in O+Cu central collisions at AGS vs central collisions defined by zero spectators (ZDC)central collisions defined by zero spectators (ZDC)

Correlations due to to B-E don’t vanishCorrelations due to to B-E don’t vanish

PRC 52, 2663 (1995)

• No studies yet at RHIC. Also centrality cut not as good at collider


k(k() linear with non-zero intercept in ) linear with non-zero intercept in p+p and Light Ion reactions.p+p and Light Ion reactions.

• This killed “intermittency” but dont ask, see E802 PRC52,2663 (1995)

Als

o se

e M

JT P

LB

347

, 431

(199

5)


Charged particle number fluctuationsCharged particle number fluctuations

“Particle number fluctuations in a canonical ensemble” V.V. Begun et al, PRC70, 034901 (2004) NA49-BariConf-JPConf 5 (2005) 74

-

+

AllN

BD

Bin

omia

lPoisson

•This is the right way to do it but more work is needed!


But Net-Charge fluctuations are studied InsteadBut Net-Charge fluctuations are studied Instead

• I really dislike net charge fluctuations compared to -,+, all.

• Because net-charge Q=N+ - N- is conserved. You have to do some work to make it fluctuate--distribute the net charge on small intervals

• But then you just get binomial statistics:

• To make matters worse, ok interesting, a theorist who obviously never took a statistics course proposed to study the variable R=n+/n-

• However, statisticians NEVER take <1/n->, which is divergent if there is any finite probability, no matter how infinitesimal, that n-=0. This is especially dumb since you have to go to small p (n-=N-p0) to get some flucuations.

• See e.g. the work of our chairman for further details.J. Nystrand, E. Stenlund, H. Tydesjo, PRC 68, 034902 (2003)


The idea of net charge fluctuations as a QGP The idea of net charge fluctuations as a QGP signature didn’t worksignature didn’t work

• The idea was that fractional charges represent more particles fluctuating than unit charged hadrons so that the normalized variance ~1/n should be smaller. All experiments just see the standard random binomial unit-charged hadron fluctuations, with a small effect due to correlations from resonances, e.g. ++-

NA49 PRC 70, 064903 (2004) CERES JPhysG30, S1371(2004)

PHENIX PRL89, 082301(2002)


Event-by-Event Average pEvent-by-Event Average pTT

• For events with n charged particles of transverse momentum pTi, MpT is just the sum divided by a constant and so has most of the same properties as ET distributions including being described by the convolutions of a Gamma Distribution.

• By its definition <MpT>=<pT> but you must work hard to make sure that your data has this property to <<< 1%.

• The random background is usually defined by mixed events. You must ensure that your mixed event sample is produced with exactly the same n distribution as the data events. Also no two tracks from the same event can appear in a mixed event.


p < 2

p=2

p > 2

dN/x

dx

x =

Inclusive pInclusive pTT spectra are Gamma Distributions spectra are Gamma Distributions


NA49-First Measurement of MpNA49-First Measurement of MpTT distribution distribution

NA49 Pb+Pb central measurement PLB 459, 679 (1999)

• Points=data; hist=mixed; minimal, if any, difference

• Very nice paper, gives all the relevant information


Statistics at Work--Analytical Formula for MpStatistics at Work--Analytical Formula for MpT T for statistically independent Emissionfor statistically independent Emission

It depends on the 4 semi-inclusive parameters: b, p of the pT distribution (Gamma) <n>, 1/k (NBD), which are derived from the quoted means and standard deviations of the semi-inclusive pT and multiplicity distributions. The result is in excellent agreement with the NA49 Pb+Pb central measurement PLB 459, 679 (1999)

See M.J.Tannenbaum PLB 498, 29 (2001)


0-5 % Centrality

Black Points = Data

Blue curve = Gamma distribution derived from inclusive pT spectra

It’s not a Gaussian…it’s a Gamma distribution!

“ “Average pAverage pTT Fluctuations” Fluctuations”

PHENIX

From one of Jeff Mitchell’s talks:


PHENIX MpPHENIX MpTT vs centrality vs centrality

200 GeV Au+Au 200 GeV Au+Au PRL PRL 9393, 092301 (04), 092301 (04)

• compare Data to Mixed events for random.

• Must use exactly the same n distribution for data and mixed events and match inclusive <pT> to <MpT>

• best fit of real to mixed is statistically unacceptable

• deviation expressed as:

FpT= MpTdata / MpTmixed -1 ~ few %

MpT (GeV/c)

MpT (GeV/c)


Large Improvement at Large Improvement at ssNNNN= 200 GeV = 200 GeV

Compared to Compared to ssNNNN= 130 GeV results = 130 GeV results

• 3 times larger solid angle• better tracking• more statistics

sNN=130 GeV PRC 66 024901 (2002)

PRL PRL 9393, 092301 (2004), 092301 (2004)


Fluctuation is a few percent of Fluctuation is a few percent of MpMpT T ::

Interesting variation with N Interesting variation with Npartpart and p and pTmax Tmax

n >3 0.2 < pT < 2.0 GeV/c 0.2 GeV/c < pT < pTmax

PHENIX nucl-ex/0310005 PRL PRL 9393, 092301 (2004), 092301 (2004)

Errors are totally systematic from run-run r.m.s variations


Npart and pNpart and pTTmaxmax dependences explained by jet dependences explained by jet

correlations with measured jet suppressioncorrelations with measured jet suppression

20-25% centrality

Other explanations proposed include percolation of color strings E.G.Ferreiro, et al, PRC69, 034901 (2004)


What e-by-e tells you that you don’t What e-by-e tells you that you don’t learn from the inclusive averagelearn from the inclusive average

• e-by-e averages separate classes of events with different average properties, for instance 17% of events could be all kaons, and 83% all pions---see C. Roland QM2004, e-by-e K/ consistent with random.

• A nice example I like is by R. Korus, et al, PRC 64, 054908 (2004): The temperature T~1/b varies event by event with T and T.


Assuming all fluctuations are from Assuming all fluctuations are from TT//TT Very small and relatively constant with Very small and relatively constant with ssNNNN

T/T

CERES tabulation H.Sako, et al, JPG 30, S1371 (04)

Where is the critical point?


What Have We LearnedWhat Have We Learned

• In central heavy ion collisions, the huge correlations in p-p collisions are washed out. The remaining correlations are:

Jets

Bose-Einstein correlations

• These correlations saturate the fluctuation measurements. No other sources of non-random fluctuations are observed. This puts a severe constraint on the critical fluctuations that were expected for a sharp phase transition but is consistent with the present expectation from lattice QCD that the transition is a smooth crossover.


What e-by-e tells you that you don’t What e-by-e tells you that you don’t learn from the inclusive averagelearn from the inclusive average


Specific HeatSpecific Heat

• Korus, et al, PRC 64, 054908 (2001) discuss specific heat:

n represents the measured particles while Ntot is all the particles, so n/Ntot is a simple geometrical factor for all experiments


Something New: cSomething New: cVV/T/T33

• Gavai, et al, hep-lat/0412036 call this same quantity cV/T3 and predict in “quenched QCD” at 2Tc and 3Tc that it differs significantly from the ideal gas. Can this be measured?

• In PHENIX, n/Ntot~1/20, so FpT ~ 0.33% for cV/T3~15. This may be possible if we go to low pTmax out of the region where jets contribute.


Worth TryingWorth Trying

0.2 GeV/c < pT < pTmax


Summary---Mortadella ReduxSummary---Mortadella Redux• No matter how you slice it---it’s still .... ..resonance matter for sNN=3-20 GeV


Mortadella-NYTimes 2/10/2000Mortadella-NYTimes 2/10/2000


BACKUPBACKUP


IV-Moments, Cumulants, CorrelationsIV-Moments, Cumulants, Correlations


RHIC 2-3 times more ERHIC 2-3 times more ETT than WNM but: than WNM but:


Are upper edge fluctuations random?Are upper edge fluctuations random?


cont’dcont’d

Korus Gavai


Begun-nuclth0411003Begun-nuclth0411003

I understand this 1/b~1/6 but I don’t understand the rest

critical point-onset april 2005m. j. tannenbaum 1/55 event-by-event fluctuations in heavy ion...

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