重イオン衝突における非ガウスゆらぎ
北沢 正清
（阪大）
MK, Asakawa, Ono, arXiv:1307.2978「熱場の量子論とその応用」、京大基研、2013年8月28日

Beam-Energy ScanTHadrons
Color SC0m

Beam-Energy ScanSTAR 2012

highT
lowHadrons
Color SC0m

Fluctuations
 Fluctuations reflect properties of matter.
 Enhancement near the critical point
Stephanov,Rajagopal,Shuryak(’98); Hatta,Stephanov(’02); Stephanov(’09);…

 Ratios between cumulants of conserved charges
Asakawa,Heinz,Muller(’00); Jeon, Koch(’00); Ejiri,Karsch,Redlich(’06)

 Signs of higher order cumulants
Asakawa,Ejiri,MK(’09); Friman,et al.(’11); Stephanov(’11)

Conserved Charges : Theoretical Advantage
 Definite definition for operators- as a Noether current- calculable on any theory
ex: on the latticeConserved Charges : Theoretical Advantage
 Definite definition for operators- as a Noether current- calculable on any theory
ex: on the latticeSimple thermodynamic relations- Intuitive interpretation forthe behaviors of cumulantsex:
Asakawa, Ejiri, MK, 2009

Conserved Charge Fluctuations
RBC-Bielefeld ’09

Cumulants of NB and NQ are suppressed at high T.
Asakawa, Heinz, Muller, 2000; Jeon, Koch, 2000;
Ejiri, Karsch, Redlich, 2006; Asakawa, Ejiri, MK, 2009;
Friman, et al., 2011; Stephanov, 2011

Proton # Cumulants @ STAR-BES
STAR,QM2012

Somethinginteresting??

Charge Fluctuation @ LHC
ALICE, PRL110,152301(2013)

D-measure• D ~ 3-4 Hadronic• D ~ 1 Quarkis not equilibrated at freeze-out at LHC energy!

Dh Dependence @ ALICEALICE
PRL 2013t

Dhz

rapidity window

Time Evolution of CC
DytNQzDy

Variation of a conserved charge in Dy isachieved only through diffusion.

The larger Dy, the slower diffusion

Dh Dependence @ ALICE
ALICEPRL 2013tDhz

Dh dependences of fluctuation
observables
Higher-order
cumulants asthermometer?
encode history of thehot medium!

Cumulants : HIC vs Lattice
格子QCDで得られた
ゆらぎ- m/T 関係線

HotQCD,LATTICE2013

実験の観測値＋格子m /T
不一致

化学凍結

<dNB2> and < dNp2 > @ LHC ?
should have different Dh dependence.

Baryon # cumulants are experimentally observable! MK, Asakawa, 2011;2012

<dNB2> and < dNp2 > @ LHC ?
should have different Dh dependence.

Baryon # cumulants are experimentally observable! MK, Asakawa, 2011;2012

<dNB2> and < dNp2 > @ LHC ?
should have different Dh dependence.

Baryon # cumulants are experimentally observable! MK, Asakawa, 2011;2012

<dNQ4> @ LHC ?
How does
suppression

behave as a function of Dh?
orenhancement

Three “NON”s
Physics of non-Gaussianity in heavy-ion
collision is a particular problem!

 Non-Gaussian Non-criticalNon-Gaussianitiy isirrelevant in large systems

Critical enhancement isnot observed in HIC so far Non-equilibrium

Fluctuations are notequilibrated in HIC

Hydrodynamic Fluctuations
Landau, Lifshitz, Statistical Mechaniqs II
Kapusta, Muller, Stephanov, 2012
Stephanov, Shuryak, 2001

Stochastic diffusion equation

Markov (temporary local)+continuity

Gaussian fluctuationin equilibrium

GaussianDiffusion Master Equation
Divide spatial coordinate into discrete cells
probabilityDiffusion Master Equation
Divide spatial coordinate into discrete cells
probability

Master Equation for P(n)Solve the DME exactly, and take a0 limit
No approx., ex. van Kampen’s system size expansion

Solution of DME
1stinitial

Deterministic part  diffusion equation
at long wave length (1/a<<k)Appropriate continuum limit with ga2=D

Solution of DME
1stinitial

Deterministic part  diffusion equation
at long wave length (1/a<<k)Appropriate continuum limit with ga2=D

2ndConsistent with stochastic diffusion eq.
(for smooth initial condition)
Shuryak, Stephanov, 2001

Net Charge Number
Prepare 2 species of (non-interacting) particles

Let us investigate
at freezeout time t

Time Evolution in Hadronic Phase
Hadronization (initial condition)

 Boost invariance / infinitely long system
 Local equilibration / local correlation

suppression owing tostrongly dependent on
local charge conservation hadronization mechanism

Time Evolution in Hadronic Phase

Time evolution via DME

Hadronization (initial condition)

Freezeout Boost invariance / infinitely long system
 Local equilibration / local correlation

suppression owing tostrongly dependent on
local charge conservation hadronization mechanismTotal Charge Number
In recombination model,

6 quarks
6 antiquarks



can fluctuate, while

does not.

Dh Dependence at Freezeout
Initial fluctuations:

2nd
4thparametersensitive tohadronization

Dh Dependence at Freezeout
Initial fluctuations:

2nd

4th

parametersensitive tohadronization

<dNQ4> @ LHCAssumptions

10.5• boost invariant system• small fluctuations of CC at hadronization• short correlation in hadronic stage

4th-order cumulant
will be suppressedat LHC energy!
Dh dependences encode
various information on thedynamics of HIC!

Global Charge Conservation

坂井田、26日ポスター

HICで作られたQGPは有限系
有限長さLの系でゆらぎの時間発展を解く系の長さL が大きいと、
Dh<<Lの領域は境界の影響を受けない

1.00.80.60.40.20246810• 有限体積効果は、Dh依存性から読み取ることができる• ALICEの結果は、 有限体積効果が寄与していないことを示唆

Global Charge Conservation

坂井田、26日ポスター

HICで作られたQGPは有限系
有限長さLの系でゆらぎの時間発展を解く系の長さL が大きいと、
Dh<<Lの領域は境界の影響を受けない

1.00.80.6

0.4

0.20246810• 有限体積効果は、Dh依存性から読み取ることができる• ALICEの結果は、 有限体積効果が寄与していないことを示唆Summary
Plenty of physics in
Dh dependences ofvarious cumulants

Physical meanings
of fluctuation obs.
in experiments.

Diagnosing dynamics of HIC
 history of hot medium mechanism of hadronization
 diffusion constantSummary
Plenty of physics in
Dh dependences ofvarious cumulants

Physical meanings
of fluctuation obs.
in experiments.

Diagnosing dynamics of HIC
 history of hot medium mechanism of hadronization
 diffusion constantSearch of QCD Phase Structure

Open Questions & Future Work
 Why the primordial fluctuations are observed only at LHC,and not RHIC ?
 Extract more information on each stage of fireballs using
fluctuations

 Model refinement
 Including the effects ofnonzero correlation length / relaxation time
global charge conservation
 Non Poissonian system  interaction of particles

Dh Dependence at STARSTAR, QM2012

decreases as Dh becomes larger at RHIC energy.

Chemical Reaction 1
x: # of X
a: # of A (fixed)
Master eq.:

Cumulants with fixed initial condition

initialequilibrium

Chemical Reaction 2

Higher-order
cumulants
grow slower.

Time Evolution in HIC
Quark-Gluon Plasma

Hadronization

FreezeoutHydrodynamic Fluctuations
Landau, Lifshitz, Statistical Mechaniqs II
Kapusta, Muller, Stephanov, 2012

Diffusion equation

Stochastic diffusion equation

Stochastic Forcedetermined by fluctuation-dissipation relation

Dh Dependence

Shuryak, Stephanov, 2001

 Initial condition:
 Translational invarianceinitialequilibrium

initialequilibriumNon-Gaussianity in Fluctuating Hydro?
It is impossible to directly extend the theoryof hydro fluctuations to treat higher orders.
 No a priori extension of FD relations to higher orders Theorem

Markov process＋continuous variable
Gaussian random forcecf) Gardiner, “Stochastic Methods”

Event-by-Event Analysis @ HIC
Fluctuations can be measured by e-by-e analysis in HIC.
STAR, PRL105 (2010)

DetectorNon-Gaussianity
fluctuations (correlations)

Non-Gaussianity

PLANCK : statistics insufficient to see non-Gaussianity…(2013)