Transcript "Anisotropies in momentum space in a Transport Approach"

Anisotropies in momentum space
in a Transport Approach
V. Greco
UNIVERSITY of CATANIA
INFN-LNS
Quantifying the properties of Hot QCD Matter – INT Seattle, July 2010
Information from non-equilibrium: Elliptic Flow
z
y
v2/ measures the efficiency
of the convertion of the anisotropy
from Coordinate to
Momentum space
Fourier expansion in p-space
x
y x
2
 x
2
y 2  x2
l(sr)1 | viscosity
c2s=dP/d | EoS
Hydrodynamics
Parton Cascade
2v2/
c2s= 0.6
Massless gas =3P -> c2s=1/3
More generally one can distinguish:
l=0
c2s= 1/3
dN
dN
1  2 v 2 cos( 2 )  ... 

dpT d dpT
c2s= 0.1
Measure of
P gradients
time
-Short range: collisions -> viscosity
-Long range: field interaction ->  ≠ 3P
Bhalerao
D.
Molnaret&al.,
M.PLB627(2005)
Gyulassy, NPA 697 (02)
First stage of RHIC
Hydrodynamics
No microscopic details
(mean free path l -> 0, h=0)


T
( x)  0

 
+ EoS P()



j

  B ( x)  0
Parton cascade
Parton elastic 22 interactions
(l1/sr - P=/3)
p   f  C 22
v2 saturation pattern reproduced
If v2 is very large
More harmonics needed to describe
an elliptic deformation -> v4
v4  cos( 4 ) 
p x4  6 p x2 p y2  p y4
( p x2  p y2 ) 2
To balance the minimum v4 >0 require
v4 ~ 4.4% if v2= 25%
At RHIC a finite v4 observed
for the first time !
Outline
 Results from RHIC
 bulk, jets, hadronization, heavy quarks
> motivation for a transport approach
 Cascade 2<->2 collisions at fixed h/s:
 Scaling properties of v2(pT)/x
 Link v2(pT) - h/s~0.1-0.2 and coalescence
 Large v4/(v2)2
 Transport Theory with Mean Field at fixed h/s:
 NJL chiral phase transition and v2 <> h/s
 Extension to quasiparticle models fitted to lQCD ,P
From the State of the Art > Transport
Initial Conditions
Quark-Gluon Plasma
BULK
(pT~T)
CGC (x<<1)
Gluon saturation
MINIJETS
(pT>>T,LQCD)
Hadronization
Microscopic
Mechanism
Matters!
Heavy Quarks
(mq>>T,LQCD)
From RHIC but more relevant at LHC:
 Initial Condition – “exotic” non equilibrium
 Bulk – Hydrodynamics BUT large finite viscosities (h,z)
 Minijets – perturbative QCD BUT strong Jet-Bulk “talk”
 Heavy Quarks – Brownian particle (?) BUT strongly coupled to Bulk
 Hadronization – Microscopic mechanism can modify QGP observables
 Non-equilibrium + microscopic scale are relevant in all the subfields
 A unified framework against a separate modelling can be useful
Viscous Hydrodynamics
Fx
v x
 h
Ayz
y


T   Tideal
  dissip
Relativistic Navier-Stokes (Hooke law like)
but it violates causality,
II0 order expansion needed -> Israel-Stewart
tensor based on entropy increase ∂ s >0
 th,tz two parameters appears
 df (pT) quite arbitrary
 df~ feq reduce the pT validity range
P. Romatschke, PRL99 (07)
Transport approach
p


 
   p F   m  m  p* f ( x, p )  C22  C23  ...
Free streaming
Field Interaction -> ≠3P
Collisions -> h≠0
C23 better not to show…
Discriminate short and long range interaction:
Collisions (l≠0) + Medium Interaction ( Ex. Chiral symmetry breaking)
r,T
decrease
Motivation for Transport approach
p


 
   p F   m  m  p* f ( x, p )  C22  C23  ...
Wider Range of validity in h, z, pT + microscopic level -> hadronization
l>0 Hydrodynamic limit can be derived
It is a 3+1D (viscous hydro 2+1D till now)
 No gradient expansion, full calculation
 valid also at intermediate pT - out of equilibrium
region of the modified hadronization at RHIC
 valid at high h/s > LHC
 include hadronization by coalescence+fragmentation
 CGC pT out of equilibrium impact (beyond the difference in x)
not possibile in hydrodynamics
 naturally including Bulk viscosity z
Transport >Cascade approach

p  f  I
2 2
 ...
Solved discretizing the
space in (h, x, y)a cells
Collision integral not solved with the geometrical
interpretation, but with a local stochastic sampling
Z. Xhu, C. Greiner, PRC71(04)
3x
t0
3x0
exact solutions of the
Boltzmann equation
Questions that we want to address:
 What scalings survive for a fluid at finite h/s?
 Can we constrain h/s by v2?
 Are both v2(pT) and v4 (pT) consistent with a unique h/s?
 Are v2(pT) and v4 (pT) at finite h/s consistent with Quark Number Scaling?
We simulate a constant shear viscosity
Cascade code
Relativistic Kinetic theory
h 4
p

 cost.
s 15 s tr n(4   T )

s tr ( r (r ), T )  s tr ,a 
pa
4
1
(*)
15 na (4   T ) h / s
a=cell index in the r-space
Time-Space dependent cross
section evaluated locally
The viscosity is kept
constant varying s
(different from D. Molnar arXiV:0806.0026
P. Huovinen-D. Molnar, PRC79 (2009))
A rough estimate of s(T)
Neglecting  and inserting in (*)
h
1

s 4
s  4n 
1
s tr  2
T
 P
T
2 2g 3

T
45
At T=200 MeV
str10 mb
G. Ferini et al., PLB670 (09)
V. Greco at al., PPNP 62 (09)
Analizing the
scaling of v2(pT)/x
 Is the finite h/s that causes the breaking of v2/ scaling?
 The v2 /<v2> scaling validates the ideal hydrodynamics?
Relation between x and v2 in Hydro
STAR, PRC77(08)
Bhalerao et al., PLB627(2005)
2v2/
Hydrodynamics
time
Ideal Hydrodynamics (no size scale):
v2/ scales with :
- impact parameter
- system size
Does the breaking
come from finite h/s?
Parton Cascade – without a freeze-out
v2/ and v2/<v2> as a function of pT
4h/s=1
Au+Au & Cu+Cu@200 AGeV
 Scaling for both v2/<v2> and v2/ for both Au+Au and Cu+Cu
 Agreement with PHENIX data for v2/<v2>
h/s1/4 on top to data, but… this is missleading
Experimentally…
PHENIX PRL 98, 162301 (2007)
v2(pT)/ does not scale!
v2(pT)/<v2> scales!
Note: Scaling also outside
the pT hydro region
STAR, PRC77 (2008)
Can a cascade approach
account for this?
Freeze-out is crucial !
Two kinetic freeze-out scheme
Finite lifetime for the QGP small h/s fluid!
a) collisions switched off
for <c=0.7 GeV/fm3
No f.o.
s tr 
b) h/s increases in the cross-over
region, faking the smooth
transition between the QGP and
the hadronic phase
1 p 1
15 n h / s
At 4h/s ~ 8 viscous hydrodynamics is not applicable!
Results with both freeze-out and no freeze-out
No f.o.
No f.o.
Au+Au@200 AGeV
v2/ scaling broken
v2/<v2> scaling kept!
Cascade at finite h/s + freeze-out :
 V2/ broken in a way similar to STAR data
 Agreement with PHENIX and STAR scaling of v2/<v2>
 Freeze-out + h/s lowers the v2(pT) at higher pT …
Short Reminder from coalescence…
 dNq

dNM
(
p
)
α
(
p
2
)
T
T
d 2 p

d 2 pT


T
2
3
 dNq

dN B
(
p
)
a
(
p
3
)
T
T
d 2 p

d 2 pT


T
dN q
dN q

1  2v 2q cos( 2 )
pT dpT dφ pT dpT


Molnar and Voloshin, PRL91 (03)
Fries-Nonaka-Muller-Bass, PRC68(03)
Enhancement
of v2
Quark
Number Scaling
v 2,M1(p T ) p 2v
(p T /2)
2,q

V2  T 
v 2,Bn(p T ) n3v
 2,q (p T /3)
 v2 for baryon is larger
and saturates at higher pT
Is it reasonable the v2q ~0.08
needed by
Coalescence scaling ?
v2q fitted from v2
GKL, PRC68(03)
Greco-Ko-Levai,PRC68(03)
Is it compatible with a fluid
h/s ~ 0.1-0.2 ?
Role of Reco for h/s estimate
Parton Cascade at fixed shear viscosity
Hadronic h/s included
> shape for v2(pT)
consistent with that needed
by coalescence
Agreement with Hydro at low pT
A quantitative estimate needs
an EoS with ≠ 3P :
vs2(T) < 1/3 -> v2 suppression ~ 30%
-> h/s ~ 0.1 may be in agreement
with coalesccence
 4h/s >3  too low v2(pT) at pT1.5 GeV/c even with coalescence
 4h/s =1 not small enough to get the large v2(pT) at pT2 GeV/c
without coalescence
Effect of h/s of the hadronic phase
Hydro evolution at h/s(QGP)
down to thermal f.o. > ~50%
Error in the evaluation of h/s
Uncertain hadronic h/s
is less relevant
Effect of h/s of the hadronic phase at LHC
Pb+Pb @ 5.5 ATeV , b= 8 fm |y|<1
The mixed phase
becomes irrelevant!
What about v4 ?
Relevance of time scale !
 v4 more sensitive to both h/s and f.o.
 v4(pT) at 4h/s12 could also be consistent with coalescence
 v4 generated later than v2 : more sensitive to properties at TTc
Very Large v4/(v2)2 ratio
Same Hydro with
the good dN/dpT and v2
Ratio v4/v22 not very much depending on h/s
and not on the initial eccentricity
and not on particle species
and not on impact parmeter…
See M. Luzum, C. Gombeaud, O. Ollitrault, arxiv:1004.2024
Effect of h/s(T) on the anisotropies
4h/s
V2 develops earlier at higher h/s
V4 develops later at lower h/s
-> v4/(v2)2 larger
2
1
QGP
T/Tc
2
1
Au+Au@200AGeV-b=8fm |y|<1
v4/(v2)2 ~ 0.8 signature of h/s
close to phase transition!
Effect of h/s(T) + f.o.
Hydrodynamics
Effect of finite h/s+f.o.
At LHC v4/(v2)2 large time scale …
Pb+Pb @ 5.5 ATeV , b= 8 fm |y|<1
4h/s=1
4h/s=1 + f.o.
4h/s(T) + f.o.
Only RHIC has
the right time
scale to infere the
T dependence of h/s!
Impact of the Mean Field and/or
of the Chiral phase transition
- From Cascade to Boltzmann-Vlasov Transport
- Impact of an NJL mean field dynamics
- Toward a transport calculation with a lQCD-EoS
NJL Mean Field
free gas
scalar field interaction

d3p


M (T )  m  4 gN f N c M (T ) 
1

f
(
T
)

f
(T )
3
(2 ) E p
L

Associated
Gap Equation
Two effects:
  ≠ 3p no longer a massless free gas, cs <1/3
 Chiral phase transition
gas
Boltzmann-Vlasov equation for the NJL
Self-Consistently
derived from NJL
lagrangian
Mass generation affects momenta > attractive contribution
Contribution of the NJL mean field
Simulating a constant h/s with a NJL mean field
Massive gas in relaxation time approximation
a=cell index in the r-space
M=0
4
h t pn
15
The viscosity is kept modifying
locally the cross-section
Dynamical evolution with NJL
Au+Au @ 200 AGeV for central collision, b=0 fm.
Does the NJL chiral phase transition affect
the elliptic flow of a fluid at fixed h/s?
S. Plumari et al., PLB689(2010)
- NJL mean field reduce the v2 : attractive field
- If h/s is fixed effect of NJL compensated by cross section increase
- v2 <> h/s not modified by NJL mean field dynamics
Extension to realistic EoS > quasiparticle model fitted to lQCD
Next step - use a quasiparticle model
with a realistic EoS [vs(T)]
for a quantitative estimate of h/s
to compare with Hydro…
but still missing the 3-body collisions
and also hadronization…
Using the QP-model of Heinz-Levai
U.Heinz and P. Levai, PRC (1998)
WB=0 guarantees
Thermodynamicaly consistency
M(T), B(T) fitted to lQCD [A. Bazavov et al. 0903.4379 ]data on  and P

NJL
P
° A. Bazavov et al. 0903.4379 hep-lat
QP
Summary
Transport at finite h/s+ f.o. can pave the way for a
consistency among known v2,4 properties:
 breaking of v2(pT)/ & persistence of v2(pT)/<v2>
scaling
 Large v4/(v2)2 ratio signature of h/s(T) (at RHIC)
 v2(pT), v4(pT) at h/s~0.1-0.2 can agrees with what needed
by coalescence (QNS)
 NJL chiral phase transition do not modify v2 <> h/s
Next Steps :
 Include the effect of an EoS fitted to lQCD
 Implement a Coalescence + Fragmentation mechanism
Simulating a constant h/s with a NJL mean field
Massive gas in relaxation time approximation
a=cell index in the r-space
M=0
h
4
t pn
15
Theory
Code
s =10 mb
The viscosity is kept modifying
locally the cross-section
Picking-up four main results at RHIC
 Nearly Perfect Fluid, Large Collective Flows:
 Hydrodynamics good describes dN/dpT + v2(pT) with mass ordering
 BUT VISCOSITY EFFECTS SIGNIFICANT
 High Opacity, Strong Jet-quenching:
 RAA(pT) <<1 flat in pT - Angular correlation triggered by jets pt<4 GeV
 STRONG BULK-JET TALK: Hydro+Jet model non applicable at pt<8-10 GeV
 Hadronization modified, Coalescence:
 B/M anomalous ratio + v2(pT) quark number scaling (QNS)
 MICROSCOPIC MECHANISM: NO Hydro+Statistical hadronization
 Heavy quarks strongly interacting:
 small RAA large v2 (hard to get both) pQCD fails: large scattering rates
 NO BROWNIAN MOTION, NO FULL THERMALIZATION ->Transport Regime
Test in a Box at equilibrium
Calculation for Au+Au running …
Boltzmann-Vlasov equation for the NJL
Numerical solution with d-function test particles
Contribution of the NJL mean field
Test in a Box with equilibrium f distribution