Glauber、 粒子多重度、中心度、揺らぎなど

益井

宙

チュートリアル研究会「重イオン衝突の物理：基礎から最先端まで」
2015年3月25-27日、理研

H. Masui / Univ. of Tsukuba

1 /42

Outline• Glauber model, multiplicity and centrality‣ Introduction of Glauber model in heavy ion collisions- more specifically, “Wounded nucleon model”‣ How to determine the centrality in experiment ?

- multiplicity model, Negative Binomial Distribution• Fluctuation‣ Why do we measure fluctuations in heavy ion collisions ?
‣ Experimental results: particle ratio fluctuation (K/π), higher moments• Summary

H. Masui / Univ. of Tsukuba

2 /42Introduction of Monte
Carlo (MC) Glauber model

H. Masui / Univ. of Tsukuba

3 /42

Glauber model
Glauber Modeling in Nuclear Collisions
M. L. Miller et al,arXiv:nucl-ex/0701025••√
The simplest
approach
to describe
theevent
initial(Au+Au
condition
Figure4: GlauberMonte Carloat ofsNN = 200 G
nucleus-nucleus
collisions
parameterb = 6 fm) viewed in the transverse plane (left panel

beamaxis (right panel).
Theandnucleons
are conditions
drawn with a rad
Widely used
to determine
centrality,
for initial
Darker disks
represent
participating
nucleons.
in hydrodynamical
models,event
generators
H. Masui / Univ. of Tsukuba

4 /42Monte Carlo (MC) Glauber model• Basic assumptions‣ nucleons travel on straight line trajectories‣ independent binary nucleon-nucleon collisions‣ inelastic nucleon-nucleon cross section is independent of number ofbinary collisions of a nucleon underwent before• Impact parameter is randomly sampled (dN/db ~ b)
• Nucleons are randomly distributed inside nuclei• Collision occurred based on the transverse distance between
nucleons, and on the measured nucleon-nucleon inelastic
cross sections (from PDG)

• Model provides impact parameter (b), number of participants(Npart), number of binary collisions (Ncoll or Nbin), and theircorrelations‣ also provides spatial anisotropy, so called “eccentricities”
H. Masui / Univ. of Tsukuba

5 /42How many parameters in Glauber model ?

•Nucleons‣ Density distribution of heavy nucleus

1⇢(r) /
1 + exp ((rR)/d)

is parameterized by Woods-saxon form (2)- radius of nucleus R, skin depth (or diffuseness parameter) d‣ Deformed nucleus needs additional parameters (1, 2, or maybe 3)- Au nucleus is deformed, Pb is spherical‣ Separation between two nucleons in a nucleus (1 or 2)- As far as I know, this option is not implemented by default at RHIC experiments• Collision‣ Measured inelastic nucleon-nucleon cross section (1)p‣ The simplest collision profile is box type(x1 x2 )2 + (y1y2 ) 2 <- additional parameters if one use non-box like collision profile

s

inel
pp

⇡

• We need additional parameters to calculate multiplicity‣ This is the place where Negative Binomial Distribution plays a role
H. Masui / Univ. of Tsukuba

6 /42

ρ(r)

Woods-saxon distribution
R=6.38 fm

1

d=0.535 fm

0.5

0
0

5

10

15

r (fm)• Constant up to r ~ 5 fm
• ρ(r) = 1/2 at r=R• Finite probability in r>R due to the diffuseness parameter dH. Masui / Univ. of Tsukuba

7 /42Total p+p cross section (PDG)
http://pdg.lbl.gov/2014/reviews/rpp2014-rev-cross-section-plots.pdf

5

5
pp
pp
Σtot and Σtot mb

2

2

102

102

5

5
s GeV

2
2

5 101 25 102 25 103 25 104 22
5

• Total elastic cross sections are also available‣ ~42 2
mb is mostly used at RHIC

102

#

2

$

Π p
Π p
Σtot and Σtot mb

H. Masui / Univ. of Tsukuba

10

8 /42

Snapshot of 1 collision at b=6 fm
Glauber Modeling in Nuclear Collisions

10
M. L. Miller et al,arXiv:nucl-ex/0701025√
Figure 4: Glauber Monte Carlo event (Au+Au at sNN = 200 GeV with impact
parameter b = 6 fm) viewed in the transverse plane (left panel) and! along thebeam axis (right panel). The nucleons are drawn with a radius
Darker disks represent participating nucleons.

NN /π/2.
σinel• Event display of 1 event (impact parameter b=6 fm)• Positions of nucleon can be fluctuated event-by-event → N
The optical form of the Glauber theory is based on continuous nucleon density
distributions. The theory does not locate nucleons at specific spatial coordinates,asis thecase for the Monteformulation
is discussed
in the next
section.
etc
fluctuate
evenCarlo
if we
fix b thatin MC
GlaubermodelThis diﬀerence between the optical and Monte Carlo approaches can lead to subtle
diﬀerences in calculated results, as will be discussed below.

H. Masui / Univ. of Tsukuba

part

9 /42

dNevent/db

Impact parameter distribution
400

Au+Au at sNN = 200 GeV
Nevent=100 K events

300

Ncoll = 0

200

∝b

100
0
0

5

10

15

20

Impact parameter (fm)• We have collisions in b>2R because of Woods-saxon form
‣ but collision (Ncoll>0) isn’t always occurred at high b

H. Masui / Univ. of Tsukuba

10 /42

dN/dNcoll

dN/dNpart

Npart & Ncoll distributions
Au+Au at sNN = 200 GeV
Nevent=100 K events

4

10

3

10

105
104
103

102

102

10
100

200

300

400

Npart

1500

Ncoll

Npart, Ncoll

0

Npart
Ncoll

1000

10

0

500

1000

1500

Ncoll

1500
4/3

Ncoll ∝ Npart

103

1000

102
500

500

0
0

5

10

15

20

Impact parameter (fm)

• Characteristic shape of N , N• N ∝N
part

coll

0
0

10
100

200

300

400

1

Npart

coll (to be discussed later)

4/3
part

H. Masui / Univ. of Tsukuba

11 /42

Spatial
anisotropy(eccentricity)
Glauber Modeling in Nuclear Collisions"RP =reaction
plane

2
x
2 +
x
2

2
x = hx i"P P =

qstandard eccentricity(reaction plane eccentricity)

hxi ,

( x2

2
xy = hxyi

2
y
,
2
y
2

10

2
2
=
hy
i
y

2 )2 + 4( 2 )2
y
xy
2 +
x

hxihyi2yhyi2 ,,

participant eccentricity

Spatial anisotropy (eccentricity) √•Figure 4: Glauber Monte Carlo event (Au+Au at s = 200 GeV with impactNN‣ originallydefinedrespect
to thereactionplane and along theparameterb = 6 fm)viewedwith
in thetransverse
plane(left panel)!

•

NN /π/2.
σinelbeam axis(right panel).
The nucleons
are drawn
witha radius
PHOBOS
collaboration
come
up withbetterdefinition
Darker disks represent participating nucleons.
‣ takes into account the fluctuation of nucleon positions‣ “participant eccentricity” with respect to the “participant plane”
The optical form of the Glauber theory is based on continuous
nucleon density
H. Masui / Univ. of Tsukuba

12 /422 i [29]—is0.02
the relevant eccentricity for the azimuthal
h"
part
shown
in
Fig.
4.
on separation of 0.4 fm, to agree with HIJING [15]. A
anisotropy.
crucialofimportancethe definitionof eccentricity
er ofThe
sourcessystematicoferror
were studied,in-Fluctuations!
ng nuclear radius, nuclear skin depth, nucleon-in comparing Cu-Cu and Au-Au results for a given beam

on inelastic cross section !NN , and minimum nucleon
ation. The systematic error contributed by each
e was determined by varying that specific 〈parameterε 〉 Cu-Cu
part
analysis0.8
within reasonable limits and quantifying
the〈ε 〉 Cu-Cu
std
e in thefinal
eccentricity
resultasafunctionof
0.6〈ε 〉 Au-Au
partlity. The individual contributions were addedin〈ε 〉 Au-Au
ature to0.4
determine the 90% confidence levelstd errors
n in Fig. 4.
e crucial0.2
importance of the definition of eccentricity
mparing Cu-Cu
and Au-Au results for a given beam
0

std

v2

0.04

0.02

〉
part

v2/ 〈εpart

0.8〈ε 〉 Cu-Cu
std
FIG.
4.
The
averageeccentricity
definedways, (h"std i
0.6〈ε in〉 two
Au-Au
partN
and h"part i), as described in the text, versus
part for simulated
p!!!!!!!! 〈ε 〉 Au-Au
std GeV. The bands
0.4Cu-Cu and Au-Au collisions at sNN " 200show the 90% C.L. systematic errors.

00.2

participant eccentricity

00.2‣ fluctuation! 250 300 350
50
100 150 200

Npart. The average eccentricity defined in two ways, (h" i

0.1

0

1

100

200GeV Au-Au h±
62.4GeV Au-Au h±
200GeV Cu-Cu h±
62.4GeV Cu-Cu h±
200

Npart

(b)

300

part

0.3

242302-4
v2/ 〈ε2〉fluctuations• Geometric v scaling by

(c)

0.5
FIG.
5. (a) v2 (unscaled) vs Npart , (b) v2 =h"std i vs Npart , and(c) v =h" i vs Npart , for Cu-Cu and Au-Au collisions atp!!!!!!!! 2 partsNN " 62:4 and 200 GeV. 1-! statistical error bars are shown.
0.4 shown in j"j < 1 for the hit-based method.
v2 is
(c)

• Eccentricity increases by

0.2

0.4

std

Npart 〈ε 〉 Cu-Cu

v2/ 〈ε 〉50 100 150 200 250 300 350

0.5

0.3

1.5

0

1

0.06

〈ε〉

-0.2

1.5

v2/ 〈ε 〉0.08

200GeV Au-Au h±(b)
PHOBOS; PRL98, 242302(2007)±
62.4GeV Au-Au h
200GeV Cu-Cu h±(a)±62.4GeV Cu-Cu h

0.2
0.1

0

100

200

Npart

300

H. vs
Masui
FIG. 5. (a) v (unscaled)
N / Univ.
, (b)ofvTsukuba=h" i vs N 13, /42
andApplications:
(1)
R
P
H
Y
S
I CAA
AL
V
91, N
7
RAA

OLUME

UMBER

1.8

Central π0 (0-10%)
1.6

Peripheral π0 (80-92%)

1.4

the associated averaged number of binary collisions(hNcoll i) and participating nucleons (hNpart i) in each cenII).
Fully
corrected and combined PbSc
week
R Etrality
V I EbinW (Table
L
E
T
T
E
R
S
AUG
and PbGl !0 pT distributions
are shownin Fig.
115forPHENIX; PRL91,
072301(2003)
minimum bias and for nine centrality bins scaled byabsorption[7]. However, based solely on thefactorsof 10.
Wesented
quantify
the medium
on highpT production
here,
we are effects
not ableto distinguish
betw
in AA
collisions
with thenuclearfactortonic
or hadronic[13]
energymodification
loss scenarios.
given byThethe centrality
ratio of thedependence
measured AAofinvariant
the highyields
pT !0
to the NN collision scaled p " p invariant yields:sion is shown in Fig. 3 as a function of hNp
evt 2 !0
1.2
suppression is characterizedasT dythe ratio of#1=NAA$d NAA =dp
RAA #pT $ %
;
(1)
0
inel
2
!
over p ! p yields
integrated
above
4
GeV=c
and
hN
i=#
&
d
#
=dp
dy
coll
pp
T
pp
1
ized using two
different scalings. RAA (circles)
inel
part
where the hNcoll i=#pp is just the average Glauber nuclear
theNscaling
as
in
Eq.
(1),
whereas
R
coll
0.8
AA
overlap function, hTAuAu i, in the centrality bin underindicates Npart scaling expected in scenarios do
consideration (Table
II). RAA #pT $ measures the deviation
0.6
eitherbygluon
saturation superposition[8] or by surfaceof AA data from an incoherentof NNemthe quenched jets [10]. Figure 3 indicates that
0.4
sition from the Ncoll scaling behavior (RAA # 1)
072301-4
in the most peripheral region, to the strong su
0.2
seen in central reactions (RAA # 0:25) is smoothtion, although there is no exact participant
0
0
2
4
6
8
10
0
(Rpart>1
forallcentralities),
the!
producAApT (GeV/c)participant pair above 4 GeV=c is approxima
stant over a wide range of intermediate centra
FIG. 2. Nuclear modification factor RAA %pT & for !0 in central
Nuclearmodification
factorhigh pT agreement with a parton saturatio(closed circles)
and peripheral(open circles)
Au ! RAuAAat at qualitative
p!!!!!!!!
prediction [8].
sNN $ 200 GeV. The error bars include all point-to-point‣ TestRAA
1 →bands
A+Arepis superposition
p+p momentum spectra oexperimental(p !Np,collAuscaling;! Au) errors.
The=shaded
In summary,oftransverse
resent the fractional uncertainties in hTAuAu i and in the !0pions have been measured at midrapidity up‣ Any deviations
from Rwhich
1→
information
at early
stageof collisions
AA =canyields normalization
added in quadrature,moveall
10 GeV=c
for nine
centrality
bins of Au ! Au c!!!!!!!!
p
the points up or down together (in the central case the shaded
at sNN $ 200 GeV. The spectral shape and•band shown is the fractional error for the first point).

H. Masui / Univ. of Tsukuba

14 /42

Applications: (2) flow, asHBT

s95p-PCE [75, 76]. InitialE895
parameters
the models were cal-UrQMD(-0.6<y<0.6,in
7.4-29.7%)
0.4
7.7 GeV
ibrated using measured multiplicity
distributions(and extrap-(2+1)D hydro EOS-Q
CERES (-1<y<-0.5,
10-25%)
0.25
hydro EOS-H
STAR
10-30%)
olations
and(-0.5<y<0.5,
to describe
pT -spectra and v(2+1)D-0.5<y<0.520.35 to lower energies)
0.2
STAR(-1<y<-0.5,
10-30%)
measurements for 200 GeV
Au+Au
collisions
at RHIC. The(2+1)D hydro EOS-I(2+1)D hydro MC-KLN
0.15
STARsimilar(0.5<y<1,lifetimes,
10-30%)
two cases
but in the MC-(2+1)D hydro MC-GLB
0.3 were found to yield
0.1
KLN case the initial eccentricities are larger (more out-of0.05Au+Au π±-π±plane0.25
extended). The MC-KLN model achieves a less round π±-π±kT = 0.15-0.6 GeV/cshape simply because it starts with larger initial eccentricity
00.2
0.3 0.3
0.3[74]. The excitation function for freeze-out eccentricities has
11.5 GeV
39 GeVthe potential
to resolve ambiguities between models with dif0.25 0.25
0.25
0.15
ferent initial conditions and values of η/s. In particular, the0.2 0.2
0.2
two sets
0.1 of initial conditions and η/s used here yield identi0.15 0.15
0.15
cal v2 , but very different εF . So the results in Fig. 27 provide
0.1 0.1
0.1
0.05
tighter
constraints on these models.
0.05 0.05
0.05
The 0goal of [74] was to map systematic trends in observ0 00ables with the two models, not to explain the data precisely.
3
2
10
10
10
0.3 0.3
0.3
sNN [GeV]
In fact, the applicability of these models is known to be prob19.6 GeV
62.4 GeV
0.25 0.25
0.25
lematic at lower energies both because they assume boostFIG.
27:(Coloronline) The
dependenceof the kinetic
freeze-out
eccentricity
of pions
on collisionenergy in mid-central Au+Au co
0.2 0.2
0.2
invariance,
whichisbrokenat
lower
energy,andbecausethe(E895, STAR) and Pb+Au collisions (CERES) for three rapidity regions and with ⟨kT ⟩ ≈ 0.31 GeV/c. For clarity, the points for forw
0.15 0.15
0.15
isoffset
expected
becomemore
importantat lower
backward hadronic
rapidity fromphase
STAR areslightly. toErrorbars include
only statisticaluncertainties.
Several (2+1)D hydrodynamical mod
UrQMD calculations
areshown.
Model
centralities
correspond
to
the
data.
The
trend
is
consistentcollision energy. A more realistic calculation requires (3+1)Dwith a monotonic decrease in ecce
0.1 0.1
0.1
with beam energy.
‣ hydrodynamics. Nevertheless, the new calculations
viscous0.05 0.05
0.05
are able to match more closely the experimental results. Of
0 0
0.1 0.2 0.3 0.4
0.1 0.2 0.3 0.4 0 the eccentricities at kinetic freeze-out in Fig. 27 were madeglobal tofit method
hasalbeen developed.
the hydrodynamical models, MC-GLB is closest
the data
0.3 0.3
0.3
with UrQMD in cascade mode and so do not incorporate thisA wide variety of HBT measurements have been perf
27 GeV
200 GeVthoughit still
overpredicts the freeze-out eccentricity
and thepotential
betweenstringfragments.
andthe comparison of results at different energies is0.25 0.25
0.25
slope
relevant
fromis
It should
be appears
noted thattoo
nonesteep.
of theOne
modelspredict observationall
improved.
In the[74]azimuthally
integrated case, the beam
0.2 0.2
0.2
observables
simultaneously.
The
UrQMD
model,
whileitdependence ofthe radii
that in these models the decrease in the eccentricity
with
en-generally agree with results frommatchesthefreeze-out
shapeswell,
matchesthemomentum
experiments,
butshow
a much smoother trend than the
0.15 0.15
0.15
ergy appears
to Andbe duemainly to models,
an extended
lifetime
ratherspace observables
less well.
the hydrodynamic
data which were extracted from a variety of experimen
0.1 0.1
0.1
thanlargeranisotropyof pressure
As discussed
while they
are ableto describe
the momentum
space pTgradients.
specvariationsin analysis at
techniques. The current analyse
tra and vthe
results,
do
less
well
at
predicting
theeccentricity
2
tionally
contributedata
in previously unexplored regi
end
of
Sec.
VI
A,
the
lifetime
extracted
fromRvalueslong
0.05 0.05
0.05‣
and trends observed in HBT analyses. The availability ofcollisionenergy. Thethetransverse mass dependence is als
suggestresults
an increase
the total
lifetime.
However,experimental
provide aninimportant
oppor0 0
0 these newalso
sistent with earlier observations and allows one to coin bins exhibit similar trends as afurthercannot
constrainallow
models.one to determine whether
thattheall kTdecrease
and centrality
0.1 0.2 0.3 0.4 0
0.1 0.2 0.3 0.4 tunity to dataεPPtion ofenergy.
eccentricity is due solely to increased lifetime
orcollision
whethertheenergy dependence of the volume of the hompressure gradients may also play a significant
H. Masui The/role.
Univ.
of Tsukuba
15 /42
VII. CONCLUSIONS
0.3

εF

0.25
0.2
0.15
0.1
0.05
0
0
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.3
0.25

0.2
0.15
0.1
0.05
00
0
0.3
0.25
0.2
0.15

0.1

0.2

0.3

0.4

0.1

0.2

0.3

0.4

0

Global Fit

0.1

0.2

εF

0.3

23
STAR; arXiv:1403.4972v1[nucl-ex]

0.3

0.4

• Eccentricity at freeze-out

0.1

0.2

0.3

0.4

0

0.1

0.2

0.3

0.4

measured by using azimuthal
sensitive HBT• Information for time evolution inheavy ion collisions

0.1
0.05
0
0

by comparing initial & freeze-out
eccentricity

Possible improvements• Proton distribution‣ consider point-like nucleons by default• Effect of neutrons‣ Inelastic cross section ?
‣ Radius of nucleus (and perhaps skin depth as well)• Adjustment of radius of nuclei‣ relevant if one starts considering the nucleon distributions inside

nuclei‣ deformation also affects•…
‣ NOTE: Comments above might not be relevant for experiments at

LHC. Some of experiments might have already considered these kindof effects
H. Masui / Univ. of Tsukuba

16 /42

Centrality determination

H. Masui / Univ. of Tsukuba

17 /42

Centrality determination• Centrality

centrality (%) ⌘

✓

1
tot

◆

⇥ 100

‣ Fraction of events in terms of total geometrical cross section- 0% at most central, 100% at most peripheral‣ Impact parameter cannot be measured experimentally• Basic assumption is monotonic relationship between impact
parameter and multiplicity‣ Multiplicity monotonically decrease with b• Centrality is determined by various ways (detectors)‣ the TPC at midrapidity (STAR)‣ the BBC (and/or the ZDC) at forward rapidity (PHENIX)‣ the V0 counter (ALICE)• They essentially measure charged particles‣ not the case for the ZDC, and for the FCAL in ATLAS
H. Masui / Univ. of Tsukuba

18 /42and various implementations of centrality definition.

Centrality
determination(cartoon)
3.1 Methodology
M. L. Miller et al,arXiv:nucl-ex/0701025

Figure 8: A cartoon example of the correlation of the final state observable
Nch with Glauber calculated quantities (b, Npart ). The plotted distribution andvarious values are illustrative and not actual measurements (T. Ullrich, private
communication).
H. Masui / Univ. of Tsukuba

19 /42

How to model multiplicity distribution ?

• Two componentmodel has been widely useddN
= µ (1d⌘

Npartx)+ xNcoll
2‣ particles from initial hard scattering carry some fraction (x) of

produced particles, x is O(0.1)‣ parameter µ controls the overall scale (or mean) of multiplicity• PHENIX uses simple N

part scaling with power α

dN
=µ
d⌘

✓

Npart
2

◆↵

• Multiplicity can be calculable once N

part (and Ncoll) values are

obtained by MC Glauber model‣ Is this good enough to reproduce multiplicity in experiments ?
‣ The answer is no. Let’s take a look at the result
H. Masui / Univ. of Tsukuba

20 /42

Need additional fluctuations• Underestimate the tail even by using simple poisson
fluctuation (compare black with red)

‣ If one doesn’t consider any additional fluctuations (not shown here),then results will be even worse, i.e. lower than red curve
H. Masui / Univ. of Tsukuba

21 /42

Basic idea

• Independent emission source‣ assume particles will be produced independently from each source- “source” would be participant pair, number of binary collisions or their mixture‣ mean particle number is determined by multiplicity models in previous

slide‣ use some PDF (gaussian, NBD etc) to add fluctuations for number of

produced particles‣ Negative Binomial Distribution (NBD) is mostly used to take into

account additional fluctuations• Tune parameters in PDF (and multiplicity model) to reproduce
the experimental data

‣ “Fit” the data by multiplicity model + PDF

H. Masui / Univ. of Tsukuba

22 /42

to nearlyover andachanges fromnearly exponential
distributions
and NBDs.
The Gaussian⟨kc ⟩ (dead)⟨kc ⟩ (fake)small range dependsof pseudorapidity,
less
than
one
unit.
on the position of the window and the average
It shouldmultiplicity
notedthatinthea exact
detailsof the centralitybeselected
centralitybin, respectively. By
cut are important for the shapes of the distributions in Fig. 4
treating these contributions as independent systematic error
and for the analysis by moments. The shapes of the distributotalaresystematicδ⟨k ⟩ (total) on ⟨kc ⟩ in eachvalue
determinederror
tions below sources,
the meantheby the ccentrality
was usedcut. For the present PHENIX;
to defineanalyses, the ZCAL
PRC76,
034903(2007)

same number of events, 19 667. The solid lines on the figure
are to help guide the eye to the data points from the same
6y bin however, they also represent NBD fits to the distributions which will be discussed, extensively, in due course.
The average multiplicities in the subintervals (n(8'r/)) increase rather linearly with the widths of the subintervals,
that PRC52,
the pseudorapidity
indicating
density of the average
E-802;
2663 (1995)

Negative Binomial Distribution
—

I

I

I

I

t

I

I

I

I

I

I

I

I

I

I

I

l

I

I

1

I

I

I

I

107

I
I

104

7

1/8 × 10 (δη=0.087)

6

10

/c

6

2/8 × 10 (δη=0.175)

5

10

tral

5

3/8 × 10 (δη=0.263)

4

10

4

4/8 × 10 (δη=0.350)

3

102

10

FIG. 4. Multiplicity distributions from Fig. 22for theBrg intervals 0. 1,0.2, 0.3,0.5. 1.0, as indi10
P(n)bg =0. 1

1oo

10Therefore, it is g
multiplicity distrfunction via the k
As a demonst
Fig. 3 shows the3

5/8 × 10 (δη=0.438)
6/8 × 10 2 (δη=0.525)

cated. The data for each interval are plotted
10scaled in multiplicity by (n), the mean multiplic1ity in the interval. Each successive distribution
has been normalized by the factor indicated, for-1
10 clarity of presentation.

1

7/8 × 10 (δη=0.613)
8/8 (δη=0.700)

10-2
-3

10

8g =0.5
x 10

1O-4

I

0

I

I

I

I

1

I

I

I

I

I

3

2
n//

(n)

4

5

-4

10

-5

10

0

0.5

1

1.5

2
n/<n>

2.5

3

3.5

4

• Charged particle multiplicity distribution is empiricallydescribed well by Negative Binomial Distribution‣

in A+A, p+p, and e+ + e-

P (n; µ, k) =(n + k)(n 1) (k)✓µ/k
1 + µ/k• Poisson (k→∞), Binomial (k<0), Bose-Einstein (k=1)

034903-8◆n1(1 + µ/k)k‣ k reflects the degree of correlation among particles
H. Masui / Univ. of Tsukuba

23 /42

Construct multiplicity• Demonstration with arbitrary parameters• Multiplicity distribution‣ peak at peripheral, (relatively) flat region, tail at the most central- mostly driven by linear impact parameter dependence- additional NBD fluctuation increase the ‘width’ (tail)H. Masui / Univ. of Tsukuba

24 /42

What else ?

• Acceptance & tracking efficiency (particle-wise)• Trigger (in)efficiency (event-wise)‣ we miss peripheral events, where number of produced particles issmall so that trigger counters cannot observe any particles‣ this effect would be visible in the reduction of peripheral peak on

multiplicity distribution• Auto-correlation (or self-correlation)‣ correlation between centrality and physics observables if one

determine the centrality at the same detector(s) which we carry out
the analysis- see, e.g. PHOBOS white paper, Nucl. Phys. A757, 28 (2005)‣ significant effects on fluctuation observables- even at the most central (0-5%) collisions•…

H. Masui / Univ. of Tsukuba

25 /42

Fluctuations

H. Masui / Univ. of Tsukuba

26 /42

Why do we measure fluctuations ?

• Good tool to study QCD phase diagram‣ information about the properties of the system (e.g. d.o.f.)‣ key signature for phase transition; susceptibilities (fluctuation) diverge

at 2nd order phase transition- prominent example is CMB by COBE, WMAP, … → constrain important parameters forour universe• Ultimate goal(s) in heavy ion collisions‣ search for QCD critical point (and 1st order phase transition)➡ Beam Energy Scan (BES), vary baryon chemical potential- extensive studies at SPS- recent RHIC BES phase-I- future RHIC BES phase-II, CBM at FAIR, J-PARC, …

• In this talk, focus on experimental results for K/π fluctuation,and higher moments for conserved charges

H. Masui / Univ. of Tsukuba

27 /42

Strangeness enhancement, K/π fluctuation• Proposed by J. Rafelski and R. Hagedorn (1981)

Statistical Mechanics of Quarks and Hadrons,
Edited by H. Satz @ North-Holland Publishing
Company, p253-272, 1981

• s/q → K/π etc as signatures of QGP• Study K/π ratio fluctuation to search for the phase transition- “The measurement of K/π fluctuations by NA49 collaboration were the first event-by-

event fluctuations measurement in heavy-ion experiment” (V. Koch, arXiv:0810.2520v1[nucl-th])• Now, K/π fluctuation is also used to search for the QCD
critical point‣ though it is not clear (to me) K/π fluctuation is really sensitive to CEP
H. Masui / Univ. of Tsukuba

28 /42Observable
hNK (NK 1)i hN⇡ (N⇡ 1)i⌫dyn,K⇡ =+hNK i2hN⇡ i2hNK N⇡ i2
hNK ihN⇡ i• ν = 0 for the lack of dynamical correlation• NA49 uses different definition σ
dyn

dyn

‣ see e.g. PRC79, 044910 (2009)1
1- σdyn is the same with νdyn if statistical fluctuation ⌫stat = hNK i + hN⇡ iis small• Advantage of νdyn

‣ insensitive to efficiency corrections (factorial moment)‣ no mixed events
H. Masui / Univ. of Tsukuba

29 /42

NA49 vs STAR•

Excitation Function for νdyn,K/π
NA49 σdyn,K/π converted to νdyn,K/π using σ2dyn = νdyn.

•

TPC+TOF (GeV/c):• π : 0.2 < pT < 1.4• K : 0.2 < pT < 1.4•

TPC+TOF includes
statistical and systematic
errors from electron
contamination.
• Pion contamination of ka< 3% using TPC and TOF.

•

Difference between STA
and NA49 result below √= 11.5 GeV.

(NA49 data from C. Alt et al. [NACollab.], Phys. Rev. C 79, 04491(2009)•

Both models show little
acceptance effects.

• UrQMD predicts little endependence.
• HSD predicts an energy
dependence.
13• NA49 shows rapid increase with decreasing beam energy• STAR shows constant down to 7.7 GeV• Discrepancy between NA49 and STAR
Terence Tarnowsky

DNP 2011
October 26-29. 2011‣ not resolved yet‣ acceptance ? particle identification ?
H. Masui / Univ. of Tsukuba

30 /42

Search for QCD critical point
http://www.bnl.gov/bnlweb/pubaf/pr/photos/2012/07/RHIC_Graphics_Fig1-HR.jpg• QCD critical point searchis one of the main goals inheavy-ion experiments• Theoretical approach(lattice QCD) is valid insmall µB at this point‣ probably valid up to µB/T ~ 1
critical point ?
First orderphase transition ?

➡ experimental search• In order to explore the

QCD phase diagram, weneed to vary baryon
density (baryon chemical
potential)➡ Beam Energy Scan
H. Masui / Univ. of Tsukuba

31 /42Observables ?

• What is the best observable to search for the QCD criticalpoint ? → Fluctuation !

• Why ?

‣ correlation length and susceptibilities diverge at critical point‣ but they are not direct observables in experiments• What are actual observables ?

‣ Moments (cumulants) of conserved charges (e.g. net-baryons)‣ Before RHIC BES, we mostly focused on 2nd moment (width)• Why conserved charges ?

‣ Direct connection to susceptibilities (calculable in lattice QCD)⌦

2 = ( N )
3
S =
⇠
2

2

↵

2

⌦

3

↵

⇠ ⇠ , 3 = ( N ) ⇠ ⇠
4
3
4
2
, K =
⇠
2
2
2

4.5

⌦

, 4 = ( N )

4

↵

2

3 h( N )i ⇠ ⇠ 7

H. Masui / Univ. of Tsukuba

32 /42Observables ?
⌦2 = ( N )
3
S =
⇠
2

2

↵

2

⌦

3

↵

⇠ ⇠ , 3 = ( N ) ⇠ ⇠
4
3
4
2
, K =
⇠
2
2
2

4.5

⌦

, 4 = ( N )

4

↵

2

3 h( N )i ⇠ ⇠ 7• Higher order moments (cumulants) are more sensitive tocorrelation length (see power)• Product of moments (ratio of cumulants) ↔ ratio ofsusceptibilities‣ by taking the ratio volume effect is canceled out (good for experimentsince we cannot measure volume of the system)• What is the signal of critical point ?

‣ Non-monotonic energy dependence of the product of moments (ratioof cumulants) for conserved charges

H. Masui / Univ. of Tsukuba

33 /42

Fluctuation of “conserved” charge ?
cartoon from
Kitazawa-san’s slideV
Detector• Fluctuation should be 0 if we are able to measure all particles• Measure event-by-event fluctuation in the (limited) detector
acceptance‣ in pseudorapidity range ±O(1)H. Masui / Univ. of Tsukuba

34 /42

Non-gaussian fluctuation
From Wikipedia• 3rd moment = skewness‣ asymmetry• 4th moment = kurtosis‣ peakedness• Both moments = 0 for gaussian distribution• Critical point search → non-gaussian fluctuations
H. Masui / Univ. of Tsukuba

35 /42Baseline
Skellamdistribution
Poisson & Skellam distributionsk, (k = 0, 1, 2, · · · )k!
✓ ◆k/2µ1p(µ1 +µ2 )Skellam distribution : p(k) = e
Ik (2 µ1 µ2 ), (k = · · · , 2, 1, 0, 1, 2, · · · )
µ2
µ1 µ2
1
mean = µ1 µ2 , variance = µ1 + µ2 , skewness =, kurtosis =3/2µ1 + µ2(µ1 + µ2 )
µ1 µ2
!S =
,  2=1(µ1 + µ2 )Poisson distribution : p(k) =eSkellam distribution•• Whatwould be the baseline we compare with ?
‣ Difference of two statistically independent random variables each

‣ Skellam
distribution- difference
two statisticallyhaving Poisson
distribution
with ofdifferent
expectedindependent
values µ and µ1

random variables, each having Poisson distribution with different
Net-proton(protonminus
expected valuesµ1 andµ2 anti-proton) distribution would

2•
follow Skellam distribution
and anti-particle distributions are Poisson, then the•‣ Ififparticle
proton & anti-proton
distributions net-protons
are Poisson distributiondifferenceof them (net-charge,etc) follow
distribution
Direct derivationof moments is tough for Skellam distribution• Skellam‣‣ productof even(or odd)
order
cumulants willbe 1 is very powerful for
You would
see that
the useof generating
function

this particular example
H. Masui / Univ. of Tsukuba

36 /42

Net-proton fluctuations
STAR: PRL112, 032302 (2014)

PRL 112, 032302 (2014)
1.2
1.0

Au+Au Collisions at RHIC
Skellam Distribution

0.8

Sσ

PHYSICAL REVIEW LETTERS

70%-80%
0%-5%

0.6
0.4

Net proton

0.2

0.4<pT<0.8 (GeV/c),|y|<0.5

1.2

κ σ2

• Latest published results• Compare with the baseline 1

the κσ 2 and Sσ from UrQMD [23] (not shown in the figures)closely follows the data at the lower beam energies of 7.7
and 11.5 GeV. Their values are, in general, larger compared(middle& bottompanels)to data
for the higher
beam energies.
The current data provide the most relevant measurements
aroundoverInteresting
the widest range structurein μB (20–450 MeV)
to date for the
CP search, and for comparison with the baryon number
19.6 GeV
susceptibilities computed from QCD to understand thevarious features of the QCD phase structure [6,16,17]. The
A lot of experimental
deviations of Sσ and κσ 2 below the Skellam expectation
are developments
qualitatively consistent with
QCD-based
model whichto areachhere
includes a CP [33]. However, the UrQMD model which‣ auto-correlationdoes
not include a CP alsoeffectshows deviations from the
Skellam expectation. Hence, conclusions on the existence
statisticalerror
calculationof‣ CPcan be madeonly
after comparison to QCD
calculations with CP behavior which include the dynamics‣ efficiency corrections
associated with heavy-ion collisions, such as finite correlation length and freeze-out effects.
CaveatIn summary, measurements of the higher moments and2
theirproducts(Sσandκσ) of the net-proton distributions‣ Net-protons ≠ net-baryons
at midrapidity (jyj < 0.5) within 0.4 < pT < 0.8 GeV=c in
pﬃﬃﬃﬃﬃﬃﬃﬃ
þ Au collisions
over a wide
range024904
of sNNand μB have
M.Au
Kitazawa,
M. Asakawa;
PRC86,(2012),
PRC86,
069902 to(2012)
been presented
search for a possible CP and signals of aphase transition in the collisions. These observables show acentrality and energy dependence, which are not repro-

•
•

1.0
0.8

p+p data
Au+Au 70%-80%
Au+Au 0%-5%
Au+Au 0%-5% (UrQMD)
Ind. Prod. (0-5%)

0.6
0.4
1.05

(S σ)/Skellam

week ending
24 JANUARY 2014

1.00
0.95

•

0.90
0.85
5 6 7 8 10

20

30 40

100

200

Colliding Energy sNN (GeV)

FIG. 4 (color online). Collision energy and centrality dependence of the net proton Sσ and κσ 2 from Au þ Au and p þ pcollisions at RHIC. Crosses, open squares, and filled circles

H. Masui / Univ. of Tsukuba

37 /42

measurements for a CP in the region of the phase planewith "B < 200 MeV. It is difficult to rule out the existenceof CP for the entire "B region below 200 MeV. The extent

History of net-proton fluctuations
STAR: PRL105, 022302 (2010)µB (MeV)
720

420

STAR Data
AMPT

κ σ2

2

AMPT(SM)

210

54

20

10

Au+Au Collisions
0.4<pT<0.8 (GeV/c)|y|<0.5

HIJINGTHERMINATOR
URQMD

1Future Critical Point Search

3 4 56

10

20 30

100

200

sNN (GeV)
pﬃﬃﬃﬃﬃﬃﬃﬃ
2 for net-proton
FIG. 4 (color online). firstsNN dependenceof#!
resultdistributions measured at RHIC. The results are compared topﬃﬃﬃﬃﬃﬃﬃﬃ
non-CP model calculations (slightly shifted in sNN ). The leftright arrow at the bottom indicates the energy range for the CP
search at RHIC.

STAR preliminary

first BES result (QM2010)

2-5

H. Masui / Univ. of Tsukuba

38 /42

History of net-proton fluctuations

PHYSICAL RE
PRL 112, 032302(2014)
Moment Products: Energy Dependence!
STAR: PRL112, 032302 (2014)second BES result (QM2012)

1.2
1.0

Au+Au Collisions at RHIC

SσSkellam Distribution! Deviations
0.8 below Poisson !
70%-80%expectations
0.6 are observed beyond
0%-5%statistical andsystematic
errorsin
0.4
Net proton
2
0-5% most0.2central
collisionsforκσ0.4<pT<0.8 (GeV/c),|y|<0.5
and Sσ above 7.7 GeV.
1.2

!
κ σ2

1.0
! For peripheral
collisions, the !
deviations0.8
above Poission expectations
p+p data
are observed below 19.6 GeV.!
Au+Au 70%-80%
0.6!

(S σ)/Skellam! UrQMD0.4model show monotonic !
behavior 1.05
for the moment products, in!
which non-CP
physics, such as!
1.00
baryon conservation, hadronic
0.95
scattering effects, are implemented.!

Au+Au 0%-5%
Au+Au 0%-5% (UrQMD)
Ind. Prod. (0-5%)

0.90
0.85
5 6 7 8 10

20

30 40

100

200

Colliding Energy sNN (GeV)• Independent centrality determination, correct statistical error

FIG. 4 (color online). Collision energy and centrality dependence of the net proton Sσ and κσ 2 from Au þ Au and p þ pcollisions at RHIC. Crosses, open squares, and filled circles
are for the efficiency corrected results of p þ p, 70%–80%, and
0%–5% Au þ Au collisions,
Skellam distributions
H. Masuirespectively.
/ Univ. of Tsukuba
39 /42

calculation (+ efficiency correction for final results)

Historyof net-proton
fluctuations
EnergyDependenceof Cumulants RatiosSTAR: CPOD2014
STAR PreliminarySTAR Preliminary

centrality
determination,correct
statisticalerror•1)Independent
Error bars are
statistical
only. Systematic
errorsestimation•

calculation,
underway.efficiency
Dominantcorrection
contributors: a) efficiency correctionsb) PID;What
has been changed from “published” results ?

‣ pTNon-monotonic
cut: 0.4 < pT < 0.8
GeV/c (TPC)→ 0.4 < patT <2 GeV/c(TPC+TOF)
2)
behavior
is observedthemost central
collisions (0-5%).
H. Masui / Univ. of Tsukuba

40 /42

So what ?

• We need differential study in phase space (p , η)T

‣ We don’t know what would be the optimal window of pT, η‣ Naively, focus on low pT (bulk) would be better, while we miss ~50%

of protons with pT < 1 GeV/c cut off

Results: Mean Net-p, p an

• Efficiency correction is important

Results: Me‣ for both cumulants and their statistical error- Lower efficiency with the TOF → correction factor is larger

STA• We need to measure conserved charge‣ net-charge is better ?

‣ neutrons → Hadron calorimeter (J-PARC ?)- we essentially measure protons only at low beam energies! Mean Net-proton, proton and anti-proton number in! Net-proton number is dominated by protons at low
increases when energy decreases(Interplay between baryon stopping and pair produ
H. Masui / Univ. of Tsukuba
41 /42

Summary• MC Glauber model is convenient tool (for experimentalists) tostudy initial conditions, determine centrality, and relate it withinitial geometry (Npart etc)

‣ Possible improvements can be done to make the centrality

determination to be more precise• Fluctuation observables are important to search for the QCD
critical point• Future experiments should provide precise measurements
below 20 GeV‣ Future BES phase-II, starting from 2018 or 2019, CBM at FAIR, J-

PARC- neutron detection would provide more precise measurements on net-baryon
fluctuations at low energies

H. Masui / Univ. of Tsukuba

42 /42