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Relativistic nuclear collision Relativistic nuclear collision in pQCD and corresponding in pQCD and corresponding

dynamic simulationdynamic simulation

Ben-Hao Sa

China Institute of Atomic Energy

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• INTRODUCTION• HADRON-HADRON COLLI. IN pQCD• DYNAMIC SIMULATION FOR hh COLLI. (PYTHIA MODEL)• NUCLEUS-NUCLEUS COLLI. IN pQCD• DYNAMIC SIMULATION FOR NUCLEU

S-NUCLEUS COLLI. (PACIAE MODEL)• LONGITUDINAL SCALING

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INTRODUCTION

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RHIC, hottest physical frontier in particle

and nuclear physics

• Primary goal of RHIC:

Study properties of extremely high

energy and high density matter• Explore phase transition from HM to

QGM, QGP transition• Evidences for sQGP, existed, however,

it is still debated

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• The ways studying RHIC: Perturbative QCD (pQCD) Phenomenologic model (eg. NJL) Hydrodynamic Dynamical simulation:

– Hadron cascade model:

PYTHIA,RQMD,HIJING,VENUS,

QGSM, HSD, LUCIAE

(JPCIAE), AMPT, uRQMD, etc.

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– Parton and hadron cascade model: PCM (VNI), AMPT (string melting), PACIAE Zhe Xu & C. Greiner

– Better parton and hadron cascade model, required by present RHIC experiments

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HADRON-HADRON

COLLI. IN pQCD

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1. Cross section of hadron production

in hadron-hadron ( + ) colli.

hh colli. = superposition of

parton-parton colli.

a b

min min

,3,

1 12 2 2

/ /

1( ; , )

( , ) ( , ) ( , )i j

T cmi j

hi j i a i j b j k k

x x

dE a b h x s p

d p

dx dx f x Q f x Q D z Q

1

( ; , )k

dij kl s t

z dt

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: parton ( ) distribution function in

hadron ( )

: momentum fraction taking by

from

: scale of scattering

: fragmentation function of to

/i af

ix

2 24 TQ p

hkD

1 2k

i j

x xz

x x

i

a

i

a

k h

( ; , )d

ij kl s tdt

: cross section of sub-process

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2. cross section of partonic sub-process

1 2

min min 21

2 1

1 1,

2 2

2 , tan( )2

,1

TT h

h

cmTT h

ii j

i

xu tx x x

s s

px

sx xx

x xx x x

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Subprocess cross section expresses as

(after average and sum over initial

and final states)

LO pQCD cross section of seven

contributed processes and two processes

with photon are:

2 22

2 2( : , ) | ( ) |s s

ij kl

dij kl s t ij kl

dt s s

M

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2 2 2 2 2 2 2

2 2 2

2 2 2 2 2 2 2 2 2

2 2 2

2 2 2

4 4 8, ( ) ,

9 9 27

4 8 32 8( ) , ,

9 27 27 3

1 3

6 8

i j i j i j i j i i i i

i i i i i i

i i

q q q q q q q q q q q q

q q q q q q gg

gg q q

s u s u s t s

t t u ut

s u t u u t u t u

t s ts ut s

t u t u

ut

2 2 2 2 2

2 2

2 2 2

2 2

4, ,

99

(3 );2

8 1( ), ( )

9 3

i i

i i i i

q g q g

gg gg

em emq q g i q g q i

s s

s u s u

s us tut us st

s t uu t t s

e et u s t

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Mandelstum variables:

2 2 2

1 2 1 3 1 4( ) , ( ) , ( )s p p t p p u p p

p3

p2 p4

p1

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3. Parton distribution function (PDF) and

fragmentation (decay) function (PFF)

• Can’t calculate from first principle• There are a lot of parameterizations

based on the experimental data

of lepton-hadron deep inelastic

scatterings (for PDF) and/or of the

annihilations (for PFF)

e e

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• Most simplest PDF (without depen.) at large x region is something like

• Most simple PFF is some thing like

• Total fractional momentum carried by :

034 (1 )

5g

zD

z

2 , TQ p

3 4 10

7 8

( ) ~ (1 ) , ( ) ~ (1 ) , ( ) ~ (1 ) ,

( ) ~ (1 ) , ( ) ( ) 0.1(1 )

xu x x xd x x xu x x

xd x x s x s x x

, ,q q g

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• Approximately 3/5 of parton momentum goes to pions and the rest to kaon and baryon pair.• As gluon is a flavor isosinglet its momentum equally distributes among

0, ,

1

arg

0

/

1

/

0

[ ( ) ( ) ( )

( ) ( ) ( )] 0.5 ( ( ) ( ))

( ) 0.5

ch e

u p

g p

x dxx u x d x s x

u x d x s x u x f x

dxxf x

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DYNAMIC SIMULATION FOR

HADRON-HADRON COLLI.

(PYTHIA MODEL)

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Remnant

Remnant Initial state radiation

Final state radiation

Hadronization

kf

lf

Rescattering ?

Decay

)(xf pi

)(xf pj

h

p

…

ˆd

dt

• Sketch for pp simulation in PYTHIA

Parton distributionfunction

p

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• Differences from pQCD are:

– Monte Carlo simulation instead

of analytic calculation

– There is additions of initial and

final states QCD radiations

– String fragmentation instead of rule

played by fragmentation function in

pQCD

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– Semihard interactions between other partons of two incoming hadrons (multiple interaction)– Addition of soft QCD process such as diffractive, elastic, and non-diffractive (minimum-bias event)– Remnant may have a net color charge to relate to the rest of final state– Multiple string fragmentation

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0q

0q

0q

0q

1q

1q

0q

10qq

1q2q2q

Multiple String Fragmentation

…

– Hadron rescattering (?) and decay

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We have expended PYTHIA 6.4 including

parton scattering and then hadronization

(both string fragmentation and coalecense)

and hadron rescattering. We are please if

you are interested to use it

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NUCLEUS-NUCLEUS COLLI. IN pQCD

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A) Hadron production cross section in

nucleus-nucleus (A+B) collision is

calculated under assumptions of

– Nucleus-nucleus collision is a

superposition of nucleon-nucleon

collision

– A+B reaction system is assumed to be a continuous medium

B) Convolution method

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bB

b-bB+bA

bA

b

oA

oB

A

B

``Skecth of AB collision projected to

transverse plane”

(beam, i. e. z axis, is perpendicular to page)

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3 3

3

( , ) ( , )

( ) ( )

( ) ( , ), ( ) 1

AB pp

h A A A A A B B B B A B hh h

pp

A B A A B B A hh

A A A A A A A A A

d dE db dz b z db dz b b b z E

d p d p

ddb db b b b b E

d p

b dz b z b db

–The cross section can be expressed as

: normalized thickness function of nucleus

– Phenomenological considerations for:

( )A Ab

A

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• Nuclear shadowing

• Multiple scattering (ela. diffractive,…)

• Jet quenching (energy lose)

3 3( , ) ( , )

(...) (...) (...)

AB pp

h A A A A A B B B B A B hh h

d dE db dz b z db dz b b b z E

d p d p

S M Q

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C) Glauber method

(Glauber theory with nn inelastic cross

replaced by pQCD nn cross section)

– : probability having a nn colli.( )t b db

within transverse area when nucleonpasses at impact parameter : thickness function of nn collision

db

a

c b

c

bdb

( )t b

nucleon a

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– : probability finding a nucleon in volume in nucleus A at , which is normalized as

– Probability for occurring an inela. nn colli. when nucleus A passes B at an im

pact parameter is

A Adb dz

( , )A Ab z

( , )A A A A Ab z d b d z

( ) 1 ( 1)t b db total probability

( , ) 1AA A A Adb dz b z

b

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– Probability for occurring n inela. colli. is

as there can be up to collisions– Total probability for occurring an event

( , ) ( , ) ( )

( ) ( ) ( ) ( )

( ) : . ( ) 1

A A A A A B B B B B A B in

A B A A B B A B in in

b z db dz b z db dz t b b b

db db b b t b b b T b

T b thickness functionof A B colli T b db

( , ) (1 ) , ( )n AB nin

ABp n b s s s T b

n

A B

combinations Probability having an inela. Colli.

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of nucleus-nucleus inela. colli. at impact parameter is

– Total cross section of above event is

– If one use pQCD p+p cross section instead of in above equations one has pQCD inela. cross section for (A+B) colli.

b

1

( , ) 1 (1 )AB

ABAB

n

dp n b s

db

[1 (1 ) ]ABAB db s

in

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DYNAMIC SIMULATION OF

NUCLEUS-NUCLEUS COLLI.

(PACIAE MODEL)

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Overview for PACIAE model:– In PACIAE model

• Nucleus-nucleus colli. is decomposed into nucleon-nucleon (nn) colli.

• nn colli. is described by PYTHIA, where nn colli. is decomposed into parton-parto

n colli. described by pQCD– The PACIAE constructs a huge building us

ing block of PYTHIA & plays a role like convolution in nucleus-nucleus cooli. in pQCD

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– The PACIAE model is composed of

(1) Parton initialization

(2) Parton evolution

(3) Hadronization

(4) Hadron evolution

four parts

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(1) Parton initialization Nucleon in colliding nucleus is

distributed due to Woods-Saxon ( ) and

4 (solid angel) distributions Nucleon is given beam momentum Nucleon moves along straight line nn collision happens if their least approa

ching distence

r

mintotd

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their collision time is then calculated Particle (nucleon) list

order # of particle four momenta

. .

. .

. .

and nn collision time list

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order # of colliding pair collis. time

. . .

. . .

. . .

are constructed

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A nn collision with least colli. time, selected in colli. time list, executed by PYTHIA with fragmentation switched off Consequence of nn collision is a configuration of and g ( if diquark (anti-diquark) is forced splitting into randomly) Nucleon propagate along straight line in time interval equal to difference between last and current colli. times

( )q q

( )qq qq

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Update particle list, i. e. move out colliding particles and put in produced particles Update colli. (time) list:

Move out colli. pairs which constituent involves colliding particle Add colli. pairs with components one from colliding nucleon and another from

particle list Next nn colli. is selected in updated colli. list, processes above are repeated until nn colli. list is empty

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(2) Parton evolution (scattering)

• Only 2→2 process, considered for parton scattering and LO pQCD cross section,employed.

• If LO pQCD differential cross section denotes as

2

ij kl s

ij kl

d

dt s

s pa®

®

= å

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• For process of , for instance

• That has to be regularized as

by introducing color screening mass

1 2 1 2q q q q®

2121

22

22

9

4

qqqq t

us

2121

2

22

9

4

qqqq t

us

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• Total cross section of sub-process

(4)

at high energy

• Using above cross sections parton scattering can be simulated by MC

td

dtds klij

lksij

,

0

ˆˆ

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(3) Hadronization

• Partons begin to hadronize when their interactions have ceased (freeze-out).

• Hadronized by:

— Fragmentation model : Field-Feynman model (IF) Lund siring fragmentation model

— Coalescence model

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• Ingredients of coalescence model: Field-Feynman parton generation me

chanism is applied to deexcite energetic parton and increase parton multiplicity like multiple fragmentation of string in Lund modelThe gluons are forcibly splitting into

pair randomlyThere is a hadron table composed of

qq

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Field-Feynman parton

generation mechanism

1q0q 2q 3q 4q 5q2q 3q 4q5q1q

Original quark jet Created quark pairs from vacuum

…

1 3 5 7 9 11 …

(if mother with enough energy)

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mesons & baryons, made of u, d, s, & c quarks Meson: pseudoscalar and vector mesons, and Baryon: SU(4) multiplets of baryons and Two partons, coalesce into a meson, three

partons into a baryon (anti-baryon), due to their flavor, momentum, and spatial coordinate and according to valence quark

0bL

g

B

00 *, , ,B B B

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structure of hadron If coalescing partons can form either a p

seudoscalar or a vector meson, such as can form either a or a ,

a principle of less discrepancy between invariant mass of coalescing partons and mass of coalesced hadron invoked to select one from them The same for baryon.Three momentum conservation is require

d

ud p+r +

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Phase space requirement 2 3

3 316

9

hr p

g

pD D =

3

:

g 4:

r:

Volume occupied by a single hadron

in phase space

Spin and parity degeneracies

relative distance between coalescing partons

for meson

relative momentum between cp oal: c es i

h

g

=

D

D ng

partons for meson

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(4) Hadron evolution (rescattering) Consider only rescattering among

FOR simplicity, is assumed

at high energ Assume

Usual tow-body collision model, employed

( ), , , , , , , , , 'p n k Jp r w yL S D Y

85.0totinel

nnnpnpp

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LOGITUDINAL SCALING

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• Longitudinal scaling (rapidity scaling):– Eg. ,

independent to beam energy– A kind of limiting fragmentation ansatz (1969)

– First observed by BRAHMS (2001), then PHOBOS (2003-2005) – Using PACIAE to confront with that

• Model parameters are fixed, except b in Lund string fragmentation, b is assumed approximately proportion to

h h h h= -' '/ ( ),ch beamdN d y

NNs

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• Results

– Charged particle transverse momentum distribution

.

exp.

5001 3977 2788

200 1 30 62.4 19.

5060 250 4170 210 2845 142 16

6

1

6 5 2 0.

80 1

588

12

00

ch

NN

theo

chN

s eV

b

N

G

± ± ± ±

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h h/ ( )chdN d– The

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– Longitudinal (rapidity) scaling

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• v2 longitudinal scaling

– v2 together with jet quenching is an

evidence of sQGP

– Important and widely studied observable,

did not well introduced

– Give a exact deduction starting from

invariant cross section as follows s

µ3 3

3 3

d d NE Ed p d p

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Transferring into momentum cylindrical system,

substituting pz by y, and using

we have density function

=/ 1 /z

dy dp E

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If distribution function N can separated

then multiplicity density function reads

where superscript on N is omitted

If proper normalization is introduced as follow

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the study of v2(y) should be started from

multiplicity azimuthal density function

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If above density function is isotropic then

above azimuthal density function reads

if is periodic and even function, above

density function can be expended as

2p

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or

(1)

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It is obvious, < > means an average first over

particles in an event and then average over

all events if multiple events are generated .

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If azimuthal density distribution is isotropic then

because of

above azimuthal density function reduced to

so the anisoptropic effects are in

rather than

1p

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The basic paper (PR, C58(1998)1671) starts (2)

and then gave a statement

Reavtion plane: impact parameter vector in

px –py plane and pz axis : measured with respect to reaction plane

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Reaction angle : angle between reaction

plane and px axis, introduced for extracting

elliptic flow in experiment

In theory the impact parameter vector can be

fixed at px axis, so

reaction plane is just the px –pz plane

reaction angle =0 is consistent with the definition before Eq. (2), different from (1) in normal. factor

and integrals over y and pT which make

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meaning of average more transparent As azimuthal density function reduces to

in isotropic azimuth, anisotropic effect is

referred by rather than by

Because, in that paper it is mentioned,

no possible, factor was abserbed

in vn

.

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Conclusions:

1. The average, < >, should be first over

particles in an event and then over events,

rather than “over all particles in all events”

THE “over all particles in all events”

without the weight of event total multiplicity

is not correct in physics.

2. Anisotropic effects should be studied by

rather then 1p

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SPECIFIC HEAT IN HM & QGM

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• Singularity behavior of specific heat,

relevant to phase transition

• Confusing status at present:Specific heat of charged pions=

60 ±100, from experimental charged pion transverse momentum distribution

in Pb+Pb colli. at 158A GeVSpecific heat=1.66, from simulated

p+

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transverse mass distribution in Pb+ Pb

colli. at 158A GeV by JPCIAE

(a hadron and string cascade model) A specific heat of 13.2 was found

for pions in a pure statistical modelQCD matter (QGM) specific heat,

found to be larger than an ideal

gas of ～ 30 in thermodynamic potential

of pQCD

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However, in a pure gauge theory,

specific heat of QGM is lower than

ideal gas of ～ 21

• To cleaning up, a parton and hadron cascade model, PACIAE, used to study specific heat

of HM (represented by ) and QGM ( ) in an unified framework

u d g

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• Heat capacity, , is the quantity of heat needed to raise the temperature of a system by one unit of temperature (e. g. one GeV)

where T, V, N, S and E are, respectively, the temperature, volume, number of particles, entropy, and internal energy of system

,v

V V N

S EC T

T T

vC

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• Specific heat, : heat capacity per particle which composes the system• Fitting the measured (calculated) particle

transverse momentum distribution

to an exponential distribution

temperature, T, extracted event-by-event

1t

t t

dNP p

p dp

( ) exp tT t

pP p A

T

vc

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• If reaction system (fireball), equilibrium, event-by-event temperature fluctuation obeys

: mean (equilibrium) temperature : temperature variance

2

2

( )( ) ~ exp[ ]

2vC T

P TT

T T T T

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• Comparing above temperature distribution

to the general Gaussian distribution

one finds following expression for heat

capacity

2

2

1 1 ( )( ) exp[ ]

22

xP x

22

2

1

v

T T

C T

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• Three kinds of simulations

– Default (complete) simulations

labeled by “HM v. QGM”

– Simulation ended at partonic scattering, labeled by “QGM”

– Pure hadronic cascade simulation

labeled by “HM ”

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Transverse momentum distribution of HM ( ) and QGM ( ) systems are sum of their constituents with weight of their multiplicity

t t tHM

M MP p P p P p

M M M M

gu dt t t tQG M u d g

u d g u d g u d g

MM MP p P p P p P p

M M M M M M M M M

u d g

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• Temperature of HM and QGM systems is obtained by fitting above transverse momentum distribution to an exponential distribution, within event-by-event, respectively

• Heat capacity of HM & QGM is obtained

• HM specific heat, for instance, reads

1tp

vv

Cc

M M

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QGM in initial partonic stage and HM infinal hadronic stage, seem to be in equilibrium

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• T , increase with• T in “HM v. QGM”>T in

“HM” reflecting effect of initial partonic state

• ,decreases with ,a measure of temperature fluctuationThe higher temperature the lower fluctuation

• in “HM v. QGM”, a bit larger than in “HM”, attributed to competition between temperature and multiplicity fluctuation

vc

NNS

vc

vc

NNS

vc

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• CONCLUSIONS

(a) HM specific heat excitation function

resulting from “HM v. QGM” simulations is

close to the one from “HM ” simulations

(b) That indicates QGM specific heat, hard

to survive the hadronization

(c) There is no peak structure in “QGM”,

“HM v. QGM”, & “HM” specific heat

excitation functions in studied energy

region

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Thank you !!!