エネルギー走査実験と	  バリオン数ゆらぎ	
北沢正清	  (Osaka	  U.)	Heavy	  Ion	  Pub,	  2011/12/16,	  Osaka	  U.	

Outline 	
1. ゆらぎを用いたQCD相構造の探索
2. バリオンおよび陽子数ゆらぎの関係について	
MK, Asakawa, arXiv:1107.2755	

Energy Scan Program @ RHIC 	
high	Tbeam energy	low	Hadrons
Color SC
0µ	

Energy Scan Program @ RHIC 	
high	Tbeam energy	low	Hadrons
Color SC
0µ	

Fluctuations 	
P(N)	
平衡状態において、物理量はゆらいでいる。	

N	V	N	

ゆらぎを特徴づける量	 Variance:	 Skewness:	 Kurtosis:	 And much higher…	

N	Event-by-Event Analysis @ HIC 	Fluctuations can be measured by e-by-e analysis in experiments.

V	
Detector	Variation of NQ in a rapidity range is small for conserved charges.	
Asakawa, et al., ’00; Jeon, Koch, ’00; Shuryak, Stephanov, ’02

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

Detector	Variation of NQ in a rapidity range is small for conserved charges.	
Asakawa, et al., ’00; Jeon, Koch, ’00; Shuryak, Stephanov, ’02

イジング模型とゆらぎ 	

N個のスピン	イジング模型とゆらぎ 	

N個のスピン	イジング模型とゆらぎ 	

N個のスピン	

N/L個の独立なスピンの集団	
長さLで大体同じ向き	
ゆらぎはLd/2で増大し、臨界点で発散する。	

Fluctuations at QCD Critical Point
Stephanov, Rajagopal, Shuryak ’98,’99

2nd order phase transition at the CP.
divergences in fluctuations of•pT distribution•freezeout T•baryon number, chage, …

Fluctuations at QCD Critical Point
Stephanov, Rajagopal, Shuryak ’98,’99

2nd order phase transition at the CP.
divergences in fluctuations of•pT distribution•freezeout T•baryon number, chage, …

Singular part in proton number fluctuations.	 Hatta, Stephanov, ’02Higher order moments has stronger ξ dep near the CP. Stephanov, ’09

準粒子の素電荷とゆらぎ
Boltzmann gas(=Poisson distribution)	

Quark-gluon:	

Hadrons:	
ppdsnnnpsudduu

Asakawa, Heinz, Muller, ’00
Jeon, Koch, ’00
Ejiri, Karsch, Redlich, ’06

準粒子の素電荷とゆらぎ

Quark-gluon:	

Hadrons:	
ppdsnnnpsudduu

Baryon Number 4th/2nd

Ejiri, Karsch, Redlich, ’05

Ratios between higher order moments (cumulants)
RBC-Bielefeld ’09

4th/2nd at µ =0 reflects thecharge of quasi-particles.
For,Hadrons:1

実験結果： c4/c2~1
STAR ’10,’11

Quarks:1/32

保存電荷の高次ゆらぎ 1 	
有限温度の期待値	

保存電荷の高次ゆらぎ 1 	
有限温度の期待値	

保存電荷の高次ゆらぎ 2 	

保存電荷の高次ゆらぎ 2 	

QCD相図上でのバリオン数ゆらぎ	Asakawa, Ejiri, MK, 2009	

Impact of Negative Third Moments

Once negative m3(BBB) is established, it is evidences that(1) χB has a peak structure in the QCD phase diagram.
(2) Hot matter beyond the peak is created in the collisions.
•No dependence on any specific models.
•Just the sign! No normalization (such as by Nch).

Proton # Fluctuations @ STAR 	STAR, 2010	

Proton # Fluctuations @ STAR 	STAR, 2010	

STAR, 2011 (Quark Matter)	

high µ	low µ	Proton # Fluctuations @ STAR 	STAR, 2011 (Quark Matter)	

HRG:
Hadron Resonance Gas	=	Free gas composed of
all hadrons & resonances	
Poisson distribution for
hadrons since 	

high µ	low µ	

観測にかかるゆらぎは、いつ形成されたのか？ 	
ゆらぎのダイナミクス（動的振る舞い）の議論が必要	

Detector	観測にかかるゆらぎは、いつ形成されたのか？ 	
ゆらぎのダイナミクス（動的振る舞い）の議論が必要	
保存電荷の場合	

非保存電荷の場合	

V	

V	

境界を通過する電荷のみが変化に寄与	
for	

体積内の任意の場所で電荷が変化できる	
for	

観測にかかるゆらぎは、いつ形成されたのか？ 	
Δη内の保存電荷量は、初期段階のものが終状態まで生き残ることが期待できる。	
Asakawa, Heinz, Muller, ’00
Jeon, Koch, ’00
Shuryak, Stephanov, ’02

Note:	NQΔyΔyt	ＳＴＡＲ	z	?	
 In equilibrated free nucleon gas,	 If the medium is not equilibrated,	Baryon Number Fluctuations are Better 	 than proton’s since it is not a conserved charge	
 simple theoretical treatment for conserved charges

 possible slow diffusion in hadronic stage	
Asakawa, Heintz, Muller, 2000; Jeon, Koch, 2000	

Baryon Number Fluctuations are Better 	 than proton’s since it is not a conserved charge	
 simple theoretical treatment for conserved charges

 possible slow diffusion in hadronic stage	
Asakawa, Heintz, Muller, 2000; Jeon, Koch, 2000	

 than electric charge fluctuations	 additional nonsingular contribution in Q	
Example:	
singular nonsingular	Variation of Proton # in Hadronic Phase 	 Proton number varies even after chemical freezeout
via charge exchange reactions mediated by Δ(1232):	

cross section	

200mb=20fm2	Δ(1232) 	 cross sections of p	

2	

1:2	

1	

2:1	

3	

decay rates of Δ	Δ(1232) 	 cross sections of p	2	

1:2	

1	

2:1	

3	

decay rates of Δ	Δ(1232) 	 cross sections of p	2	

1:2	

1	

2:1	

3	

decay rates of Δ	

Lifetime to create Δ+ or Δ0	c.f.) Nonaka, Bass, 2007	

BW for σ & thermal pion	Baryon & Proton Number Fluctuations 	 In general, fluctuations of NN and Np are different.
 Due to the isospin fluctuations, Np fluctuations tend to beclose to equilibrium ones than NN fluctuations.	

Slot Machine Analogy 	=+	P (N)	

N	
P (N)	N	

Examples 	
常に一定の枚数のコイン	

どの枚数も一定確率	

N	

N	

N	N	

Slot Machine Analogy 	P (N )=

P (N )B1/2(N ;N )	

：二項分布関数	Isospin Distributions 	 Large pion density	• Small nucleon density because MN/T<<1• For top RHIC energy, Nπ~20NN	
 Nucleons exclusively interact with pions	• Rare NN collisions• Huge ππ reactions	All formations and decays of Δ
take place independently	

Nonaka, Bass, ‘07	

~20fm	mesons	time	kinetic f.o.	

chem. f.o.	

hadronize	

Nucleons in Hadronic Phase 	

• rare NN collisions• no quantum corr.	

baryons	

• many pions	Probability Distribution	 	 	Detector	

 for any phase space in the final state.

binomial distribution func.	

Baryon & Proton Number Fluctuations 	 	• for isospin symmetric medium• Similar formulas up to any order!	

For free gas	Free Nucleon Gas 	Poisson distribution Pλ(N)	

Free Nucleon Gas 	
Poisson distribution Pλ(N)	

 The factrization is satisfied in free nucleon gas.	

Time Scales 	 Time scales of fireballs:	
: time scale to realize isospin binomiality 	
: time scale of baryon number diffusion	

hadronize	: life-time of hadronic medium in HIC	

Effect of Isospin Distribution 	(1)	

deviates from the equilibrium value.	

(2) Boltzmann (Poisson) distribution for 	Effect of Isospin Distribution 	(1)	

deviates from the equilibrium value.	

(2) Boltzmann (Poisson) distribution for 	

genuine info.	
For free gas	noise	Example : 3rd Moment 	

beyond the QCD phase boundary
near the CP	 Asakawa, Ejiri, MK, 2009	
Fireballs forget negative moment
over the QCD mountains?
No. Not necessarily.	
Note:Example1 : 4th/2nd 	

RBC-Bielefeld ’09~ 1 (hadron)	
~ 1/9 (quark)	
< 0 (near CP)	Ejiri, Karsch, Redlich, 2006
Stephanov, 2011	Fireballs forget primordial flucs?
No. Not necessarily.	

No suppression	Strange Baryons 	
Decay Rates:	

Decay modes:	

Regarding these ratios even, protons from these decays
is incorporated into the binomial distribution. Then, NNNB	Summary 	 Baryon and proton number fluctuations are different innonequilibrium medium. To see non-thermal contribution,
baryon number is better.
 Formulas to reveal baryon # cumulants in experiments.
 Experimental analysis of baryon # fluctuations may verify
 signals of QCD phase transition speed of baryon number diffusion in the hadronic stage.

Future Work 	 Refinement of the formulas to incorporate
nonzero isospin density / low beam-energy region
 Distribution function itself	 Discussion on time evolution of fluctuations @STAR

Nucleon Time Scales in Fireballs 	
Mean time to create Δ+,0	Freeze-out time	
Nonaka, Bass, 2007	

3rd & 4th Order Fluctuations