Fluctuation and HBT results
Kensuke HommaHiroshima University
1. Approach and probes for critical phenomena
relevant to QCD phase transitions
2. HBT
3. Two particle correlation via differential analysis onmultiplicity fluctuations
4. Summary@ATHIC08 on 15 Oct, 2008 in Tsukuba Univ., Japan
Kensuke Homma / Hiroshima Univ.

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Approach to QCD phase diagram at RHIC
¾High opacity state¾Bulk matter flow with quark d.o.f¾High temperature stateT

T=221±23(stat)±18(sys)
Lattice result Tc~170MeV

We can approach from well above TcDeconfinement p.t.(?)
Tc
Crossover mq!=0CP ?

1st order ?
Chiral p.t.
arXiv:0804.4168v1 [nucl-ex] 25 Apr 2008

μBKensuke Homma / Hiroshima Univ.

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Experimental approaches tocritical phenomena•••<qq-bar>:J/ψ suppression (deconfinment)
low mass vector mesons and dilepton continum (chiral)••••

Bulk collective observables:
Isothermal compressibility: Multiplicity fluctuations
Heat capacity: Mean pT fluctuaions
Particle ratio and fluctuations on particle compositions¾ Duration time of particle emission: HBT¾ Correlation length and the strength:
density-density correlation in longitudinal space••

Sound velocity via eccentricity scaling of v2
Viscosity to entropy ratio with v2 and RAA

Kensuke Homma / Hiroshima Univ.

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HBT as a probe of 1st order P.T.
D.H.Rischken, M. Gyulassy NPA 608(1996)479-512

Kensuke Homma / Hiroshima Univ.

√SNN (GeV)4

3-D imaging of source shape
Source function

Correlation moment
Encoding
FS interaction
Cartesian surface-sphericalharmonic basis
Alα1...αll: index from 0 to 6 here
α: spatial index

Inversion fromRl(q) to Sl(r)No assumption onsource shape
In PCMS
Kensuke Homma / Hiroshima Univ.

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3-D imaging results
PHENIX

Therminator Model¾Bjorken type longitudinal boost invariance¾Blast-wave transverse expansion¾Thermal emission from a longitudinal cylinder

PRL100,232301(2008)ρzt 2 = (τ 0 + aρ ) 2 + z 2
τ0 is proper breakup time at ρ=0
τ= τ0 + aρ is proper breakup time at ρwith a=-0.5 (burn outside in)τ is replaced by τ’ with probability ofdN/dτ’=θ(τ’−τ)/Δτ∗exp[−(τ’−τ)/Δτ]

Mean proper emission timeΔτ=2fm/c is small but finite !
(τ0~9fm/c <|ΔtLCM|>~12fm/c )Kensuke Homma / Hiroshima Univ.

6proton-proton

Low pT two particle correlation in STAR
Analyzed 1.2M minbias 200 GeV Au+Au events, and 13M 62 GeV minbias events(not shown) Included all tracks with pT > 0.15 GeV/c, |η| < 1, full φnote: 38-46% not shown

84-93%

75-84%

65-75%

55-65%

46-55%

5-9%

0-5%ΔρρrefφΔ
28-38%ηΔSTAR Preliminary
19-28%

9-19%ΔρρrefφΔ

ηΔ

We see the evolution of correlation structures fromperipheral to central Au+Au
Slide from M. Daugherity, STAR
Collaboration presented at QM08
Kensuke Homma / Hiroshima Univ.

77

Fit Function (in 5 Easy Pieces)
Proton-Proton fit functionΔρρrefSTAR PreliminaryΔρρref=φΔ

ηΔ+φΔ

ηΔΔρ
ρref

φΔ

ηΔ

Slide fromM. Daugherity, STAR
Collaboration
presented at QM08
longitudinal fragmentation
1D gaussian

HBT, res., e+e2D exponential

Minijet Peak
2D gaussian

Au-Au fit function
Use proton-proton fit function + cos(2φΔ) quadrupole term (“flow”).
This gives the simplest possible way to describe Au+Au data.
Note: from this point on we’ll include entire momentum range
instead of using soft/hard cuts
M. Daugherity, STAR Collaborationcos(2φΔ)

φΔ

dipoleAway-side-cos(φ)
quadrupole

ηΔ
88

Transition
Does the transition from narrow to broad ηΔ occur quickly or slowly?
data - fit (except same-side peak)
83-94%
STAR55-65%

inaryPrelim

STAR

inary
Prelim46-55%
STAR

0-5%

inary
PrelimSTrelim
AR P

inaryidthηΔ w

Shape changes little fromperipheral to the transition

Large change
within ~10%
centrality

Smaller change fromtransition to most central

Low-pT manifestation ofthe “ridge”

The transition occurs quickly
Slide from M. Daugherity, STAR
Collaboration presented at QM08
M. Daugherity, STAR Collaboration99

A picture of expanding medium in early stage
Initial stage

T>Tc

External field hLongitudinal fielddensity fluctuations fromthe mean density is anatural order parameterφ ( z) = ρ ( z) − ρT=Tc

Many modes from small to large appear at T=Tc.
A typical length scale disappears i.e. transition to power lawT<Tc
We may expect freeze of initially embedded fluctuation
due to rapid dilution of medium in the longitudinal direction
Kensuke Homma / Hiroshima Univ.

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Direct observable for Tc determination
GL free energy density g with φ ~ 0 from high temperature side isinsensitive to transition order, but it can be sensitive to Tc

1
1 4 1 612
2
g (T , φ , h) = g 0 − A(T )(∇φ ) + a(T )φ + bφ + cφ ⋅ ⋅ ⋅ −hφ2462
spatial correlation φ disappears at Tc → a ( T ) = a 0 ( T − T c )Fourier analysis on
G2(y)=<φ(0)φ(y)>| φk | = Y ∫ G2 ( y )e2| φ k |2 =− ik ( y )Susceptibility−11∂φk ⎛ ∂ ( g − g 0 ) ⎞⎟⎜
=
χk =
∝⎜2
⎟
a 0 (T − Tc )(1 + k 2ξ 2 )∂h ⎝ ∂φ k ⎠2dyNT
1
Y a (T ) + A(T )k 2

Susceptibility in long wavelength limitξ
1
1-D two point correlation functionNTξ (T )e −| y|/ ξ (T )G2 ( y ) = 22Y A(T )Correlation lengthξ (T ) 2 ≡A(T )a0 (T − Tc )

χ k =0 =a 0 (T − Tc )∝T

G2 (0)

Product between correlation
length and amplitude can also
be a good indicator for T~Tc
Kensuke Homma / Hiroshima Univ.

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Density measurement : dNch/dη
Cu+Cu@200GeV

Centrality

Cu+Cu@62.4GeV

P(Nch)

Negative Binomial Distribution(NBD) perfectly describes
multiplicities in all collisionsystems and centralities
at RHIC.
Nch/< Nch >p+p@200GeV
Cu+Cu@22.5GeV

Au+Au@62.4GeV

Kensuke Homma / Hiroshima Univ.

Au+Au@200GeV

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Differential multiplicity fluctuations
Δη windowΔη<0.7 integrated over Δφ<π/2 and pT>0.1GeV

Zero magnetic field toenhance low pt statistics
per collision event.
Kensuke Homma / Hiroshima Univ.

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NBD fits
at eachwindow
size in
CuCu@200

L=0Large

Level (window size)
L=28(1-δη/ΔηPHENIX)

16 fit examples inmost left edge in
top 10% events
out of 28/2*(1+28)
times NBD fits

L=240

small
Kensuke Homma / Hiroshima Univ.

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Two point correlation via NBD
Uncorrelatedsourcessource 1Correlated
sources

k=k1

k=k1

k=k1+k2

source 1+2

k=k2

k!=k1+k2

k=k2

source 2n(k )nNBD P1
Γ(n + k ) ⎛ μ / k ⎞k=1 Bose-Einstein⎜⎜⎟⎟=Γ(n − 1)Γ(k ) ⎝ 1 + μ / k ⎠ (1 + μ / k )k k=∞ Poisson

σ2 1 1= +2μμ kμ ≡< n > 1/k corresponds to integralof two point correlation

Kensuke Homma / Hiroshima Univ.

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Extraction of χk=0*T
Fit with approximated functional form
10%

Parametrization oftwo particle correlation

5%k(δη)C2 (η1, η2 ) ≡ ρ2 (η1, η2 ) − ρ1 (η1) ρ1 (η2 )
C2 (η1, η2 )−δη / ξ=α+βe2

ρ1β absorbs rapidity independent
bias: Npart fluctuation andreaction plane rotation and v2C2 (η1 ,η 2 ) = ρ 2 (η1 ,η 2 ) − ρ1 (η1 ) ρ1 (η 2 )
C2 (η1 ,η 2 ) / ρ12 ≡ αe −δη / ξ + β

Exact relation with NBD kn(n − 1)−1()
−1k δη =2n

Look atslopesδη δη

C (η ,η )dη dη∫∫=0021212δη 2 ρ12Phys. Rev. C 76, 034903 (2007)

δηApproximated k ( δη) =functional form

12 αξ / δη + β=( ξ << δη)Kensuke Homma / Hiroshima Univ.

2αξ 2 (δη / ξ − 1 + e −δη / ξ )

δη

2+β16

Correlation functions and correlation length
Using arbitrary R0, ξ and α.
Used in E802

C2 = 1 + R(0,0)e−| y1 − y2 |/ ξ1δη / 2ξk (δη) =R0 [1 − (ξ / δη)(1 − e−δη / ξ )]General correlation function

C2 = 1+k ( δη ) =R0− | y1 − y 2 | / ξe| y 1 − y 2 |αδη∫δη

R0 − y /ξedyαyOne may discuss an effective
potential form

ξ : correlation length, α : critical exponent
Kensuke Homma / Hiroshima Univ.

17αξ, β vs. Npart
Dominantly Npart fluctuations
and possibly correlation in azimuth

β is systematically shift to lower
values as the centrality bin width
becomes smaller from 10% to 5%.
This is understood as fluctuationsof Npart for given bin widths

β●5%
○10%

Au+Au@200GeVαξSTAR’s transition point●5%○10%

Phys. Rev. C 76, 034903 (2007)αξ product, which ismonotonically related with χk=0indicates the non-monotonic
behavior around Npart ~ 90.

αξ = χ k =0T / ρ1 ∝ ρ12Npart−2

T| T −T C|

Significance with Power + Gaussian:
3.98 σ (5%), 3.21 σ (10%)
Significance with Line + Gaussian:
1.24 σ (5%), 1.69 σ (10%)

Kensuke Homma / Hiroshima Univ.

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Comparison of three collision systemsαξ

Npart~90 in
AuAu@200GeV
εBJτ~2.4GeV/fm2/c

Au+Au@200GeV
Phys. Rev. C 76, 034903 (2007)Cu+Cu@200GeV<μc>/<μc>@AuAu200

Au+Au@200GeV
Phys. Rev. C 76, 034903 (2007)

Normalized mean
multiplicity to thatof top 5% in
Au+Au@200GeV

Au+Au@62.4GeV
Kensuke Homma / Hiroshima Univ.

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Similarity to STAR mini jet results at low pTPeak η Width

Peak Amplitude

Au+Au@200GeV

Au+Au200 GeV
62 GeVαξμ2X
Cu+Cu@200GeVSTAR Preliminary

εBJ

Au+Au@200GeV

Au+Au@62.4GeVSTAR Preliminary

M. Daugherity: QM2008

εBJ

Equivalent quantity;χT ∝ αξμ2 ∝ amplitude x widthshows similar trends to what
STAR sees.

<μc>/<μc>@AuAu200
Kensuke Homma / Hiroshima Univ.

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Is there other symptom?
- deviation from v2 scaling at low KET region -Npart ~90

In lower KET, there seems to be different behaviors between
baryon and mesons. The transition is at Npart~90.
Low mass sigma field may repulse pions and attract protons
according to hep-ph/0504048 by E.Shuryak .
Can this phenomenon be understood as such effects?
Kensuke Homma / Hiroshima Univ.

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How about <cc> suppression?
Au+Au@200GeV
Npart~90 inAuAu@200GeV

εBJτ~2.4GeV/fm2/c
Cu+Cu@200GeV

J/ψ suppression pattern

102
arXiv:0801.0220v1 [nucl-ex]
Kensuke Homma / Hiroshima Univ.

Npart22

Summary
1.

RHIC created strongly coupled high temperature & opaque state withpartonic d.o.f. This is the very beginning of the scientific program on
quantitative understanding of the QCD phase structure.

2.

3D imaging of source shape from HBT type correlation suggests
small but finite duration of pion emission time. There seems to be nostrong indication of 1st order P.T. at 200GeV, but worth measuring at
lower energy (in higher baryon density).

3.

PHENIX and STAR see very similar rapid transition of the twoparticle correlation in longitudinal direction at the similar centrality
range in Au+Au@200 and 62GeV. However, a caution is necessary,
because the rapidity window size is totally different: PHENIX islimited within 0.7 and perhaps STAR misses information at short
rapidity window by the brute force subtraction process. Nevertheless,
it is interesting to foresee the common reasoning.

Kensuke Homma / Hiroshima Univ.

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Open issue¾Is the rapid transition related with creation of CGC flux tube?
¾Is the color electric field in tubes related with bag pressure (confinement)?
¾Where is the threshold of the creation of the thermal system ?
CGC color flux tube

Centrality can scan AQs ∝ A

1/ 6

1/Qss~ 0.3 / 2

Figure fromK.Itakura at ISMD08

E or B, or E&B

A slide from S.H.Lee’s talk on 14 Oct

Kensuke Homma / Hiroshima Univ.

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