EventbyEvent Fluctuations in Heavy Ion Collisions

1
Event-by-Event Fluctuations in Heavy Ion
Collisions
M. J. Tannenbaum Brookhaven National Laboratory
Upton, NY 11973 USA
2nd International Workshop on the Critical Point
and Onset of Deconfinement Bergen, Norway
April 1, 2005

2
or
I dont know much about
Statistical Mechanicsbut Im really good at
Statistics!
M. J. Tannenbaum Brookhaven National Laboratory
Upton, NY 11973 USA
2nd International Workshop on the Critical Point
and Onset of Deconfinement Bergen, Norway
April 1, 2005

3
A Quick Course in Statistics

• A Statistic is a quantity computed from a sample
  (which is drawn at random from a population). 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
  

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

• Another popular statistic is the sample variance

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A 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

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Probability--II

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

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

is the standard deviation

• and the mean of the population is

• and the variance of the average is

6
Probability-III-sums?convolutions

• 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 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).
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Example-ET 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
  collision2 participants

• If ET adds independently for n collisions,
  participants, etc, the pdf is the n-fold
  convolution of f(x) p?np b?b

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NA5 (CERN) (1980) First ET dist. pp
UA1 (1982) (C.Rubbia) ?s540 GeV. No Jets
because ET is like multiplicity (n), composed of
many soft particles near !
CERN-EP-82/122. OOPS UA2 discovers jets 5 orders
of magnitude down ET distribution!
NA5 300 GeV PLB 112, 173 (1980)
2?, -0.88e) is ? dist p 2.39 0.06
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First RHI data NA35 (NA5 Calorimeter) CERN 16OPb
?sNN19.4 GeVmidrapidity
pAu is a ? dist p3.36
Upper Edge of OPb is 16 convolutions of pAu.
WPNM!!
PLB 184, 271 (1987)
WPNWounded Projectile Nucleonprojectile
participant
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E802-OAu, OCumidrapidity at AGS
?sNN5.4GeVWPNM works in detail
PLB 197, 285 (1987) ZPC 38, 35 (1988)

• Maximum energy in OCu same as OAu--Upper
  edge of OAu identical to OCu d?/dE 6
  
• Indicates large stopping at AGS 16O projectiles
  stopped in Cu so that energy emission
  (mid-rapidity) ceases
  
• Full OCu and OAu spectra described in detail
  by WPNM based on measured pAu
  

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E802-AGSMidrapidity stopping!pBe pAu have
same shape at midrapidity 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.

12
Collision Centrality MeasurementZeroDegreeCalorim
eter
PHENIX at RHIC AuAu-ZDC is biased
WA80 OAu CERN
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Extreme-Independent or 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 AA collision Npart2 Nprojpart
• Uncertainty principle and time dilation prevent
  cascading of produced particles in relativistic
  collisions ? h/mpc 10fm even at AGS energies
  particle production takes place outside the
  Nucleus in a pA 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 BA 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.

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WA80 proof of Wounded Nucleon Model at 60, 200 A
GeV using ZDC
Original Discovery by W. Busza, et al
at FNAL pA vs (Ncoll) PRD 22,
13 (1980)
PRC 44, 2736 (1991)

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ISR-BCMOR-pp,dd,?? ?sNN31GeV WNM FAILS!
WNM, AQM T.Ochiai, ZPC35,209(86)
PLB168, 158 (86)
Note WNM edge is parallel to p-p data!
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But-Gamma Dist. fits uncover Scaling in the mean
over10 decades??
p-p p2.500.06 ?-? p2.480.05
Is it Physics or a Fluke?
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Summary 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, ?sNN20 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 pA effect since QM87QM84

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i.e. The kink is a pA effect well known since
1987-seen at FNAL,ISR,AGS
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ET systematics beyond the kink

• In generic terms, dET/d? implies a measurement
  corrected for
  
• Hadronic response---correct to E-mN for baryons,
  EmN for antibaryons and E for all other
  hadrons.
  
• ET corrected to ??2?, ??1.0, scaling linearly
  in ?? x ??
  

• For fixed target dET/dydET/d?
• For collider at mid-rapidity dET/dy1.2 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)

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NA35--NA49 PbPb ?sNN17 GeV
PRL 75, 3814 (1995)
ET(2.1-3.4)-- dET/d?405 GeV_at_?sNN17 GeV
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PHENIX and E802 ET compared
?? 22.5o 2 x 22.5o 3 x 22.5o 4 x
22.5o 5 x 22.5o
E802 dET/d?128 GeV
E877 dET/d?200 GeV_at_?sNN4.8 GeV PHENIX
dET/d?606 GeV_at_?sNN200 GeV
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AuAu ET spectra at AGS and RHIC are the same
shape!!!
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dET/dy vs ?sNN for central collisions
?Bj GeV/fm3

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

• Note that ?Bj at ?sNN20 GeV is the same in OAu
  and PbPb

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ET has a dimension.Lets now consider number
distributions which are more typical of statistics
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What 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 .
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Moments 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 variancestd2)
  
• 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.

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Charged particle number fluctuations
-

All
Particle number fluctuations in a canonical
ensemble V.V. Begun et al, PRC70, 034901 (2004)
NA49-BariConf-JPConf 5 (2005) 74
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Binomial 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 q1-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 ?npconstant we get a

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Poisson 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 ?np remains constant,
  i.e. the probability of m counts when you expect
  ?.

• 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.

30
Negative 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/k?0

• 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 pp are
  Negative Binomial

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UA5--Multiplicity Distributions in (small)
intervals ?UA5 PLB 160, 193,199 (1985) 167, 476 (1986)
Distributions are Negative
Binomial, NOT POISSON implies correlations
?s540 GeV
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k vs ??2?c and ?s

• Distributions are never poisson at any ?s and ??
  
  
• Something fishy with NA49 pp result

33
NBD in OCu central collisions at AGS vs ??
central collisions defined by zero spectators
(ZDC)Correlations due to to B-E dont vanish
PRC 52, 2663 (1995)

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

34
k(??) linear with non-zero intercept in pp and
Light Ion reactions.
Also see MJT PLB 347, 431(1995)

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

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Charged particle number fluctuations
-

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

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

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But Net-Charge fluctuations are studied Instead

• I really dislike net charge fluctuations
  compared to -,, all.
  
• Because net-charge QN - 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 Rn/n-
  
• However, statisticians NEVER take , 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-p?0) 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)
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The idea of net charge fluctuations as a QGP
signature didnt 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. ????-

PHENIX PRL89, 082301(2002)
NA49 PRC 70, 064903 (2004)
CERES JPhysG30, S1371(2004)
38
Event-by-Event Average pT

• 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 but you must work
  hard to make sure that your data has this
  property to
• 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.

39
Inclusive pT spectra are Gamma Distributions
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NA49-First Measurement of MpT distribution
NA49 PbPb central measurement PLB 459, 679 (1999)

• Pointsdata histmixed minimal, if any,
  difference
  
• Very nice paper, gives all the relevant
  information
  

41
Statistics at Work--Analytical Formula for MpT
for statistically independent Emission
It depends on the 4 semi-inclusive parameters
b, p of the pT distribution (Gamma) ,
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 PbPb central measurement PLB 459, 679 (1999)
See M.J.Tannenbaum PLB 498, 29 (2001)
42
Average pT Fluctuations
From one of Jeff Mitchells talks
PHENIX
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PHENIX MpT vs centrality
200 GeV AuAu PRL 93, 092301 (04)

• compare Data to Mixed events for random.
• Must use exactly the same n distribution for
  data and mixed events and match inclusive to
  
  
• best fit of real to mixed is statistically
  unacceptable
  
• deviation expressed as
• FpT ?MpTdata / ?MpTmixed -1 few

MpT (GeV/c)
MpT (GeV/c)
44
Large Improvement at ?sNN 200 GeV Compared to
?sNN 130 GeV results
PRL 93, 092301 (2004)

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

?sNN130 GeV PRC 66 024901 (2002)
45
Fluctuation is a few percent of ?MpT
Interesting variation with Npart and pTmax
Errors are totally systematic from run-run r.m.s
variations
n 3 0.2 0.2 GeV/c PHENIX nucl-ex/0310005 PRL 93, 092301 (2004)
46
Npart and pTmax dependences explained by jet
correlations with measured jet suppression
Other explanations proposed include percolation
of color strings E.G.Ferreiro, et al, PRC69,
034901 (2004)
20-25 centrality
47
What e-by-e tells you that you dont learn 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 T1/b varies
  event by event with ?T? and ?T.

48
Assuming all fluctuations are from ?T/?T? Very
small and relatively constant with ?sNN
CERES tabulation H.Sako, et al, JPG 30, S1371
(04)
Where is the critical point?
?T/?T?
49
What 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.

50
What e-by-e tells you that you dont learn from
the inclusive average
51
Specific Heat

• Korus, et al, PRC 64, 054908 (2004) 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
52
Something New cV/T3

• 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/Ntot1/20, so FpT 0.33 for
  cV/T315. This may be possible if we go to low
  pTmax out of the region where jets contribute.

53
Worth Trying
0.2 GeV/c 54
Summary---Mortadella Redux

• No matter how you slice it---its still ....
  ..resonance matter for ?sNN3-20 GeV

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Mortadella-NYTimes 2/10/2000
56
(No Transcript)
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BACKUP
58
IV-Moments, Cumulants, Correlations
59
RHIC 2-3 times more ET than WNM but
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Are upper edge fluctuations random?
61
contd
Korus
Gavai
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Begun-nuclth0411003
I understand this 1/b1/6 but I dont understand
the rest