Initial conditions at RHIC
for hydrodynamics
Yasushi Nara(Akita International Univ.))
Tetsufumi Hirano (Univ. of Tokyo)
CGC initial conditions, KLN, fKLN, MC-KLN.
● Eccentricity fluctuations within MC-KLN● Initial fluctuations + hydrodynamics + hadronic cascade resultsfor elliptic flow in Cu+Cu and Au+Au collisions.
●

The 2nd Asian Triangle Heavy Ion Conference ATHIC 2008, Tsukuba Oct. 13-15, 2008

Hydrodynamics and inputs
Initial condition: how to start
thermalization timeenergy density(flow profile)● Equation of state and dissipative effects:
how to evolve
ideal gas EoS, lattice QCD● Freezeout: how to stop
Hadron cascade after burner●(hadronic dissipative effects)

Purpose: test uncertainties of the initial conditions.

High energy heavy ion collisions

r~1
Qs

CGC

Black box

Hydro evolution

=0 ≈1/Q s ≈1 fm /c

Hadron gas
Reaction time

Strong longitudinalcolor field -> particle

Gluon
production

Pre-equilibrium dynamics
unknownEffects of Initial condition andhadronic cascade for hydro.

Glauber + Ideal hydro + hadronic cascade underestimatesInitial transverse geometry
Glauber modeldN~ N part ,1  x ⊥  N part , 2  x ⊥ 
2d x ⊥ dy

Initial energy density or entropy is taken from
Wounded nucleon model:
number of participants or collision scaling.

Color Glass Condensate (KLN)dN g
2d x t dy=4 Nc

d 2 pt2ct

22t

2d k t  s  x 1, k  x 2,  p t −k t  ∫
2 ∫N −1
pdN
22~min{Qx,Qs ,1⊥s , 2  x ⊥ }~min {N part ,1  x ⊥  , N part , 2  x ⊥ }
2d x ⊥ dyWhy large eccentricity in KLN?
ry

CGC:

dN~min {N part ,1  x ⊥  , N part , 2  x ⊥ }
2d x ⊥ dydN~ N part ,1  x ⊥  N part , 2  x ⊥ Glauber: 2d x ⊥ dyGlauber 0, r y ~CGC 0, r y rxGlauber r x ,0CGC r x , 0

ρ glauber: ~  N part ,1 r x , 0N part , 2 r x , 0ρ CGC:~ min {N part , 1 r x , 0 , N part , 2 r x , 0}
rx

CGC Glauber

Eccentricity from KLN and fKLN

22〈 y −x 〉= 2 2〈x y 〉

fKLN: Correct treatment of surface of nucleus:
minimum saturation scale is set to the saturation scaleof nucleon.

Eccentricity fluctuations
Event-by-event fluctuations in the shape of the initial collisionzone for small systems or small transverse overlap regions
may be important.
Specifically fluctuations in the nucleon positions.
std =Ψ0Ψ0〈part 〉= 2y − 2x22

 y  x

22 22
−4 y xxy22 y  x

 2x =〈 x 2 〉−〈 x 〉2 ,  2y =〈 y 2 〉−〈 y 〉 2 ,  2xy =〈 xy 〉−〈 y 〉 〈 x 〉R.S.Hollis (PHOBOS) nucl-ex0707.0125

Eccentricity is large
MC-Glauber

ε was underestimated in early hydro calculations:
it is increased by fluctuations in the positions of nucleons
within the nucleus.

Monte-Carlo version of KLN (MC-KLN)
Sample A and B nucleons according to the Woods-Saxon distribution.
● Local density of nucleons is obtained bynumber of nucleonst A r ⊥ =S●●Saturation scale at a given transverse coordinate is given by 22 t A r ⊥  0.01
Q s , A r ⊥ =2GeV1.53x

For each generated configuration, we apply the k_t-factorization formulaat each transverse coordinate.
● Average over many events.
●

Comparison of Multiplicities
in Au+Au and Cu+Cu collisions
fKLN

Better fit when fluctuations are included.

MC-KLNEccentricity Fluctuations
from MC-KLN part =

22 22

−
4 y xxy

 2y  2x

 xy =〈 xy 〉−〈 x〉 〈 y 〉22

 x =〈 x 〉−〈 x 〉2 {2}=  〈 2part 〉 {4}=2 〈 2part 〉 2−〈 4part 〉1/ 4

event-by-event fluctuations in hydro?
Y. Hama, R. Peterson, G. Andrade, F. Grassi, W.Qian, T. Osada, C. Aguiar, T. Kodama
Initial condition for Au+Au
from NeXus

H. J. Drescher, F. M. Liu, S. Ostapchenko1, T. Pierog, and K. Werner, Phys. Rev. C 65, 054902 (2002)

Initial Condition with an Effect ofEccentricity Fluctuation
Throw a diceto choose b:
bmin<b<bmax
Rotate to realreaction planeE.g.)
bmin= 0.0fmbmax= 3.3fm
in Au+Au collisions
at 0-5% centrality

averageover events

averageover events

Initial fluctuations+Hydro+Hadron cascade
Cu+Cu

Au+Au

summary●We presented the results of hydro+hadron cascade withinitial eccentricity fluctuations within MC-Glauber and CGC(MC-KLN model).
– Eccentricity larger and elliptic flow also larger.
– Explain elliptic flow at central collisions.
–Glauber initial condition underestimates the elliptic flow
data.

–CGC initial condition slightly overestimates the data.
Main uncertainty in this work is the effect of EoSHow to implement universal
saturation scale in KLN framework?
2 A= p A Q s , n 2=Q s , n 2 A= p A Q s , n P A Probability to find at least one nucleon at agiven transverse coordinate.

 A= p A  TApA

Saturation scale for nucleon∑i≥0 p i t A i T A
T 1, A ==pA∑i≥1 p i fKLN Ansatz
dN g4  N c d 2 pt
2=dk t  s p A p B T A / p A T B / p B ∫
222 ∫d x ⊥ dy N c −1pt

≈∫
=∫dN
2
2~min {Q s , A ,Q s , B }dyd 2 pt

2dk t  p A p B T A / p A  p A p B T B / p B 2 ∫ptd2 pt
p2t2∫ d kt npart , A npart , B Recover original KLN!

Ap=1−1−T/A
If we takeANNA