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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
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M. J. Tannenbaum 1/55
or
I don’t know much about Statistical Mechanics
but I’m 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
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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:
• note that average, , is a property of the sample
 mean,
, is a property of the population
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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:
note the difference
 where
is the variance of the population
 and the mean of the population is
is the standard
deviation
 and the variance of the average is
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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=<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).
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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 collision=2 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
NA5 300 GeV PLB 112, 173 (1980)
2, -0.88<y<0.67 NO JETS!
s=23.7 GeV
Fit (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!
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First RHI data NA35 (NA5 Calorimeter)
CERN 16O+Pb sNN=19.4 GeVmidrapidity
Upper Edge of O+Pb is 16
p+Au is a  dist p=3.36
convolutions of p+Au. WPNM!!
PLB 184, 271 (1987)
WPN=Wounded Projectile Nucleon=projectile participant
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E802-O+Au, O+Cu
midrapidity at AGS
sNN=5.4GeV
WPNM works in detail
PLB 197, 285 (1987)
ZPC 38, 35 (1988)
• 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
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E802-AGS
Midrapidity 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.
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Collision Centrality Measurement
ZeroDegreeCalorimeter
participants
spectators
PHENIX at RHIC Au+Au-ZDC is biased
10-15%
5-10%
0-5%
WA80 O+Au 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 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.
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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 <n>pA vs <> (Ncoll)
PRD 22, 13 (1980)
RA= <n>pA/ <n>pp= (1+<v>) / 2
<Npart>pA
PRC 44, 2736 (1991)
= <Npart>
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<Npart>pp
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ISR-BCMOR-pp,dd, sNN=31GeV 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 p=2.50±0.06 - p=2.48±0.05
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, 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
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i.e. The kink is a p+A 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, 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)
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NA35-->NA49 Pb+Pb sNN=17 GeV
PRL 75, 3814 (1995)
ET(2.1-3.4)--> dET/d=405 GeV@sNN=17 GeV
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PHENIX and E802 ET compared
PHENIX
preliminary
 = 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@sNN=4.8 GeV PHENIX dET/d~606 GeV@sNN=200 GeV
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Au+Au ET spectra at AGS and RHIC are the same shape!!!
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Bj GeV/fm3
dET/dy vs sNN for “central collisions”
• Lines are pp s dependence. Lots of systematic issues but still kinky.
 Note that Bj at sNN=20 GeV is the same in O+Au and Pb+Pb
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ET has a dimension.
Let’s 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 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.
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Charged particle number fluctuations
Poisson
Binomial NBD
+
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 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
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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
=<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.
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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
 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
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UA5--Multiplicity Distributions in (small)
intervals ||<c around mid-rapidity are NBD
UA5 PLB 160, 193,199 (1985); 167, 476 (1986)
s=540
Distributions are Negative Binomial, NOT POISSON: implies correlations
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GeV
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k vs =2c and s
• Distributions are never poisson at any s and 
• Something fishy with NA49 p+p result
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NBD in O+Cu central collisions at AGS vs 
central collisions defined by zero spectators (ZDC)
Correlations 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
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Also see MJT PLB 347, 431(1995)
k() linear with non-zero intercept in
p+p and Light Ion reactions.
• This killed “intermittency” but dont ask, see E802 PRC52,2663 (1995)
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Charged particle number fluctuations
Poisson
Binomial NBD
+
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 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-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 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. ++-
PHENIX PRL89, 082301(2002)
NA49 PRC 70, 064903 (2004)
CERES JPhysG30, S1371(2004)
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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 <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.
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Inclusive pT spectra are Gamma Distributions
dN/x dx
p<2
p=2
p>2
x=
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NA49-First Measurement of MpT 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
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Statistics at Work--Analytical Formula for MpT
for statistically independent Emission
It depends on the 4 semiinclusive 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)
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From one of Jeff
Mitchell’s talks:
“Average pT Fluctuations”
0-5 % Centrality
PHENIX
Black Points =
Data
It’s not a
Gaussian…it’s a
Gamma distribution!
Blue curve =
Gamma
distribution
derived from
inclusive pT
spectra
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PHENIX MpT vs centrality
200 GeV Au+Au PRL 93, 092301 (04)
• compare Data to Mixed events for
random.
MpT (GeV/c)
• 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)
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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
sNN=130 GeV
PRC 66 024901 (2002)
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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 < pT < 2.0 GeV/c
0.2 GeV/c < pT < pTmax
PHENIX nucl-ex/0310005 PRL 93, 092301 (2004)
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Npart and pTmax dependences explained by jet
correlations with measured jet suppression
Other explanations proposed include
percolation of color strings E.G.Ferreiro,
20-25%
et al, PRC69, 034901 (2004)
centrality
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What e-by-e tells you that you don’t
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 T~1/b varies event by event with T and T.
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Assuming all fluctuations are from T/T
Very small and relatively constant with sNN
CERES tabulation
H.Sako, et al, JPG
30, S1371 (04)
T/T
Where is the
critical point?
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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.
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What e-by-e tells you that you don’t
learn from the inclusive average
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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
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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/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.
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Worth Trying
0.2 GeV/c < pT < pTmax
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Summary---Mortadella Redux
• No matter how you
slice it---it’s still ....
..resonance matter for
sNN=3-20 GeV
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Mortadella-NYTimes 2/10/2000
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BACKUP
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IV-Moments, Cumulants, Correlations
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RHIC 2-3 times more ET than WNM but:
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Are upper edge fluctuations random?
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cont’d
Korus
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Gavai
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Begun-nuclth0411003
I understand this 1/b~1/6 but I don’t understand the rest
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