Anisotropic flow in a Boltzmann kinetic
approach at fixed h/s(T)
V. Greco
UNIVERSITY of CATANIA
INFN-LNS
S. Plumari
M. Ruggieri
F. Scardina
nfQCD 2013 – Kyoto, 2-6 December 2013
Outline
 Transport Kinetic Theory at fixed h/s :

Motivations

How to fix locally h/s
 Two main results for HIC:
 Elliptic flow from Color Glass Condensate (fKLN)
going beyond ex and implementing also
the p-space with the Qs saturation scale

Are there hints of h/s(T) ?
Relativistic Boltzmann-Vlasov approach
{
}
p*m¶m + éë pn* F mn + m*¶m m* ùû ¶mp* f (x, p* ) = C[ f ]
Free streaming
Field Interaction (EoS)
Collisions -> h≠0
f(x,p) is the one-body distribution function
- C[feq+df] ≠ 0 deviation from ideal hydro (finite l or h/s)
- We map with C[f] the phase space evolution of a fluid at fixed h/s !
One can expand over microscopic details (2<->2,2<->3…), but in a hydro
language this is irrelevant only the global dissipative effect of C[f] is important!
Motivation for Transport approach
{
}
p*m¶m + éë pn* F mn + m*¶m m* ùû ¶mp* f (x, p* ) = C2«2 + C2«3 +...
Free streaming
Field Interaction
Collisions -> h≠0
 Starting from 1-body distribution function f(x,p) and not from Tμν:
- Include off-equilibrium at high and intermediate pT
- Implement non-equilibrium implied by CGC-Qs scale (beyond ex)
- Relevant at LHC due to large amount of minijet production
- freeze-out self-consistently related with h/s(T)
 It’s not a gradient expansion h/s:
- valid also at high h/s -> LHC (T>>Tc) or cross-over region (T≈ Tc)
 Appropriate for heavy quark dynamics (next week talk)
Simulate a fixed shear viscosity
Usually input of a transport approach are cross-sections and fields, but here we reverse
it and start from h/s with aim of creating a more direct link to viscous hydrodynamics
Chapmann-Enskog
h
s
=
p
1
1
p × th =
15
15 g( mTD )s TOT r
g(a=mD/2T) correct function that fixes the
relaxation time for the shear motion
Chapmann-Enskog
Viscosity from Green-Kubo
1
h=
T
h CE good one !
S.Plumari et al., PRC86(2012)
¥
3
xy
xy
dt
d
x
P
(x,
t)
P
(0, 0)
ò ò
0
V
Correlator estimate in a Box at fixed T
Chapman-Enskog agrees with Green-Kubo!
Simulate a fixed shear viscosity
Usually input of a transport approach are cross-sections and fields, but here we reverse
it and start from h/s with aim of creating a more direct link to viscous hydrodynamics
Chapmann-Enskog
h
s
=
p
1
1
p × th =
15
15 g( mTD )s TOT r
g(a=mD/2T) correct function that fix the
relaxation time for the shear motion
Chapman-Enskog agrees with Green-Kubo
Chapmann-Enskog
h CE good one !
S.Plumari et al., PRC86(2012)
Transport code
1 pa
1
s tot (n(r ),T ) =
15 g(a)na h / s
Space-Time dependent cross
section evaluated locally
=cell index in r-space
Viscosity fixed varying s
G. Ferini et al., PLB670 (2009)
Transport at fixed h/s vs Viscous Hydro in 1+1D
Comparison for the relaxation of pressure anisotropy PL/PT
Huovinen and Molnar, PRC79(2009)
Knudsen number-1
K
L
l


l
Large K small h/s
1 T0t 0
K0 =
5 h/s
h
1
= T ×l
s 5
In the limit of small h/s (<0.16)
transport converge to viscous hydro
at least for the evolution pL/pT
Denicol et al. have studied derivation of viscous hydro from Boltzmann kinetic theory:
PRD85 (2012) 114047
Test in 3+1D: v2/e response for almost ideal case
EoS cs2=1/3 (dN/dy tuned to RHIC)
Integrated v2 vs time
Ideal -Hydro
v2/e
Transport at h/s fixed
v2 =
p 2x - p 2y
p 2x + p 2y
Time rescaled
Bhalerao et al., PLB627(2005)
In the bulk the transport has an hydro v2/e2 response!
h/s or details of the cross section?
ds
a s2
µ
dW éq 2 (q ) + m 2 ù2
Dû
ë
cross section
Keep same h/s means:
h
s
=
1
p × th
15
t h-1 = g( mT )s TOT r
D
s TOT ( m1D ) g(m2D )
=
s TOT ( m2 D ) g(m1D )
for mD=0.7 GeV -> factor 2 larger stot
is needed respect to isotropic
h/s or details of the cross section?
ds
a s2
µ
dW éq 2 (q ) + m 2 ù2
Dû
ë
cross section
4ph/s=1
Keep same h/s means:
m h/s is really the physical parameter determining
h 1
-1
=
p × th
t h = g( Tv)satTOT
r up to 1.5-2 GeV
least
D
s
15
2
s TOT ( m1D ) g(m2D )
=
s TOT ( m2 D ) g(m1D )
 microscopic details become relevant at higher pT
 First time h/s<-> v2 hypothesis is verified!
for mD=0.7 GeV -> factor 2 larger stot
is needed respect to isotropic
h/s or details of the cross section?
ds
a s2
µ
dW éq 2 (q ) + m 2 ù2
Dû
ë
cross section
Keep same h/s means:
m h/s is really the physical parameter determining
h 1
-1
=
p × th
t h = g( Tv)satTOT
r up to 1.5-2 GeV
least
D
s
15
2
s TOT ( m1D ) g(m2D )
=
s TOT ( m2 D ) g(m1D )
 microscopic details become relevant at higher pT
 First time h/s<-> v2 hypothesis is verified!
for mD=1.4 GeV -> 25% smaller stot
for mD=5.6 GeV -> 40% smaller stot
h/s or details of the cross section?
ds
a s2
µ
dW éq 2 (q ) + m 2 ù2
Dû
ë
cross section
Keep same h/s means:
m h/s is really the physical parameter determining
h 1
-1
=
p × th
t h = g( Tv)satTOT
r up to 1.5-2 GeV
least
D
s
15
2
s TOT ( m1D ) g(m2D )
=
s TOT ( m2 D ) g(m1D )
 microscopic details become relevant at higher pT
 First time h/s<-> v2 hypothesis is verified!
Differences arises just where
in viscous hydro df becomes relevant
p  p p
df 
f eq
2
e P T
Natural extension from low to high pT
Renormalize s to fix h/s
s* (s) = Ks pQCD (s)
K(s) =1+ g e
-s/L 2
L ≈ 4-5 GeV, g fixed by h/s
pQCD limit
partons
pQCD limit, but with s=0.3
- No Q2 dependence &
- No radiative part
s=0.3 and mD=0.7 GeV
Boltzmann transport describes rise and fall of v2(pT)
Transition between low and high pT in a unified framework!
No Fine tuning! Employed the relaxation time approximation!
S. Plumari and VG, EPIC@LHC, AIP1422(2012)- arXiV:1110.4138 [hep-ph]
What about Color Glass condensate initial state?
- Kinetic Theory Qs saturation scale
z
y
x
fKLN realization of CGC
Factorization hypothesis:
convolution of parton
distribution functions
in the parent nucleus.
Kharzeev et al., PLB561, 93
(2003)
Nardi et al., PLB507, 121 (2001)
Drescher et al, PRC75, 034905
(2007)
Hirano et al., PRC79, 064904
(2009)
Albacete and Dumitru,
arXiv:1011.5161
…
Unintegrated distribution
functions (uGDFs)
x-space
p-space
dN/d2pT
Saturation scale Qs depends on:
1.) position in transverse plane;
2.) gluon rapidity.
CGC-KLN ex ≈ 30%
larger
Nardi et al., Nucl. Phys. A747, 609
(2005)than Glauber
Qsat(s)
pT
Kharzeev et al., Phys. Lett. B561, 93 (2003)
Nardi et al., Phys. Lett. B507, 121 (2001)
ex(fKLN)=0.34
Drescher and
Nara, PRC75, 034905 (2007)
exNara,
(Glaub.)=0.29
Hirano and
PRC79, 064904 (2009)
Hirano and Nara, Nucl. Phys. A743, 305 (2004)
Albacete and Dumitru, arXiv:1011.5161[hep-ph]
Albaceteet
et al.,
al., arXiv:1106.0978
T. Hirano
PLB636(06) [nucl-th]
V2 from KLN in Hydro
What does it KLN in hydro?
1) r-space from KLN (larger ex)
2) p-space thermal at t0 ≈0.6-0.9 fm/c - No Qs scale , We’ll call it fKLNTh
Larger ex - > higher h/s to get
the same v2(pT)
Luzum and Romatschke
PRC78(2008) 034915
Glauber  h/s = 0.08
CGC-KLN  h/s=0.16
See also:
Alver et al., PRC 82, 034913 (2010)
Heinz et al., PRC 83, 054910 (2011)
Implementing KLN pT distribution
AuAu@200 GeV – 20-30%
Using kinetic theory
we can implement full KLN
(x & p space) - ex=0.34, Qs =1.4 GeV
KLN only in x space ( like in Hydro)
ex=0.341, Qs=0 -> Th-KLN
Glauber in x & thermal in p
ex=0.289 , Qs=0 -> Th-Glauber
M. Ruggieri et al., Phys.Lett. B727 (2013) 177
Thermalization in less than 1 fm/c, in agreement with Greiner et al., NPA806, 287 (2008).
Not so surprising: h/s is small -> large effective scattering rate -> fast thermalization.
Longitudinal and transverse pressure
t=1/Qs≈0.1 fm/c
-> PL/PT > 0
Gelis & Epelbaum
arXiV:1307.2214
PL/PT show also a very fast equilibration (isotr≈0.3 fm/c)!
However it is not this that makes a difference for v2:
isotropization time very similar for all the cases
Longitudinal and transverse pressure
80% level of isotropization
For h/s > 0.3 one misses fast isotropization in PL/PT ( e 2 fm/c)
For h/s ≈ pQCD no isotropization
Semi-quantitative agreement with Florkowski et al., PRD88 (2013) 034028
our is 3+1D not in relax.time but full integral but no gauge field
Results with kinetic theory
Hydro - like
AuAu@200 GeV
Full x & p
M. Ruggieri et al., Phys.Lett. B727 (2013) 177 - 1303.3178 [nucl-th]
 When implementing KLN and Glauber like in Hydro we get the same of Hydro
 When implementing full KLN we get close to the data with 4ph/s =1 :
larger ex compensated by Qs saturation scale (non-equilibrium distribution)
What is going on?
We clearly see that when non-equilibrium distribution is implemented
in the initial stage (≤ 1 fm/c) v2 grows slowly respect to thermal one
Evolution with Centrality
 The difference fKLN , Th-fKLN and Th-Glauber disappears at central
collisions (like in hydro for Th-fKLN and Th-Glauber)
 In peripheral collisions fKLN would even be lower than Th-Glauber due
to non-equilibrium impact
Part II
Applying kinetic theory to A+A Collisions….
- Impact of h/s(T) on the build-up of v2(pT)
vs. beam energy
z
y
x
Terminology about freeze-out
Freeze-out is a smooth process: scattering rate < expansion rate
 h/s increases in the cross-over
region, realizing a smooth f.o. selfconsistently dependent on h/s:
 Different from hydro that is a
sudden cut of expansion at some
Tf.o.
s tr »
1 p 1
15 n h / s
h/s(T) for QCD matter
 lQCD some results for quenched
approx. (large error bars)
A. Nakamura and S. Sakai, PRL 94(2005)
H. B. Meyer, Phys. Rev. D76 (2007)
 Quasi-Particle models seem to
suggest a η/s~Tα, α ~ 1 – 1.5.
S.Plumari et al., PRD84 (2011)
M. Bluhm , Redlich, PRD (2011)
 Chiral perturbation theory (cpT)
M. Prakash et al. , Phys. Rept. 227 (1993)
J.-W. Chen et al., Phys. Rev. D76 (2007)
 Intermediate Energies – IE ( μB>T)
P. Kovtun et al.,Phys.Rev.Lett. 94 (2005) 111601.
L. P. Csernai et al., Phys.Rev.Lett. 97 (2006) 152303.
R. A. Lacey et al., Phys.Rev.Lett. 98 (2007) 092301.
W. Schmidt et al., Phys. Rev. C47, 2782 (1993)
Danielewicz et al., AIP1128, 104 (2009)
(STAR Collaboration),
arXiv:1206.5528 [nucl-
w/o minijet
Impact of h/s(T) vs √sNN
Plumari, Greco,Csernai, arXiv:1
 4πη/s=1 during all the evolution of the fireball -> no invariant v2(pT)
-> smaller v2(pT) at LHC.
 Initial minijets relevant at LHC for pT>1.5 GeV !
 LHC: almost insensitivity to cross-over (≈ 5%) : v2 from pure QGP
 Without h/s(T) increase T≤Tc we would have v2(LHC)<v2(RHIC)
Impact of h/s(T) vs √sNN
Plumari, Greco,Csernai, arXiv:1
 η/s ∝ T2 too strong T dependence→ a discrepancy about 20%.
 Invariant v2(pT) suggests a “U shape” of η/s with mild increase in QGP
 Hope: vn, n>3 with an ev.-by-ev. analysis put even stronger constraints
However for a definite statement needed hadronization + EoS-lQCD
Summary
Development of kinetic at fixed h/s(T) :
 Verified that up to ≈ 1.5-2 GeV:
v2 <-> h/s , cross section microscopic details irrelevant
 For a fluid at h/s <0.1 -> PL /PT > 0.8 at isotr <0.3 fm/c
Invariant v2(pT) from RHIC to LHC:
 Hints of the fall and rise of h/s(T) from BES?!
Sensitivity of v2 to η/s(T) anyway quite weak.
Studying the CGC (fKLN):
 Non-equilibrium implied by Qs damps v2(pT)
compensating the larger ex ->
v2(pT) can be described by 4ph/s ≈1 also for fKLN
Outlook
for the kinetic theory approach
 Hadronization:
 statistical model +Cooper-Frye vs. coalescence + fragm.
Field Dynamics:
Realistic EoS: M(T) + Bag mean field dynamics [Done!]
Include initial state fluctuations to study vn:
 more constraints on h/s(T) and initial state?
 something more or new at pT ≈2-4 GeV ?
pA …
Endeavor already undertaken….
Next step –
Include Initial State Fluctuations
(Preliminary results)
Monte Carlo Glauber
N part
{
2
2
r^ µ å exp -éë( x - xi ) + ( y - yi ) ùû / 2s 2
i=1
}
s = 0.5 fm
en =
r^n cos [ n(f - Fn )]
r^n
r^n sen [ n(f - Fn )]
1
Fn = arctan n
n
r^ cos [ n(f - Fn )]
G-Y. Qin, H. Petersen, S.A. Bass and B. Muller, PRC82,064903 (2010)
H.Holopainen, H. Niemi and K.J. Eskola, PRC83, 034901 (2011)
Initial State Fluctuations: vn vs εn (Preliminary)
4ph/s=1 ≈ 5
C(2,2)=0.93
v2
e2
v3
» 0.21
C(n, m) =
( vn -
C(3,3)=0.68
e3
vn
) ( en -
s Vn s en
en
» 0.11
)
 v2 and v3 linearly correlated to the corresponding eccentricities ε2 and ε3
Initial State Fluctuations: vn vs εn (Preliminary)
4ph/s=1 ≈ 5
C(2,2)=0.93
v2
e2
C(3,3)=0.23
» 0.21
C(n, m) =
( vn -
vn
) ( en -
s Vn s en
en
)
 v4 and ε4 weak correlated similar to hydro calculations:
F.G.Gardim,F.Grassi,M.Luzum and J.Y.Ollitrault NPA904 (2013) 503.
Niemi, Denicol, Holopainen and Huovinen PRC87(2013) 054901.
General agreement with hydro Niemi et al. PRC87(2013), but:
 h/s not constant (include cross-over increase)
- 3+1 D not 2+1D
 s =0.5 fm not 0.8 fm (if relevant at all!)
Initial State Fluctuations: vn(pT) (Preliminary)
Data taken from: A. Adare et al. [PHENIX collaboration], Phys.Rev. Lett. 107, 252301 (2011).
Like in viscous hydro the data of vn(pT) at RHIC energies are
described with 4πη/s=1 ≈ 5
Fluctuations allows to extend the studies on impact of CGC-Qs,
h/s(T) and study pA …
Distribution function & occupation number
 f >1 need of Bose-Einstein statistics
(1+f) terms needed in the collision integral
-> possibility of Bose Condensate induced by CGC
J. -P. Blaizot et al., arXiv:1305.2119 [hep-ph]; J. P. Blaizot et al., Nucl. Phys. A904-905 2013, 829c (2013).
f ≈ 1/ + slope change
 f >1 at pT> 0.5 our effect manifest at larger pT
 Longitudinal expansion lowers average density by -1
but f(p) goes down slowly at pT<0.5 GeV
 One should include also gluon to quark conversion
At LHC and low pT it could
be there and CYM helps!
Back-up
Part I
Do we really have the wanted shear viscosity h
with the relax. time approx.?
- Check h with the Green-Kubo correlator
Shear Viscosity in Box Calculation
Green-Kubo correlator
1
h=
T
microscopic details
¥
3
xy
xy
dt
d
x
P
(x,
t)
P
(0, 0)
ò ò
0
V
P xy (x,t)P xy (0, 0) = P xy (0, 0)P xy (0, 0) ×e-t/t
macroscopic
observables
4 eT
=
15 V
η ↔ σ(θ), r, M, T …. ?
S. Plumari et al., arxiv:1208.0481;see also:
Wesp et al., Phys. Rev. C 84, 054911 (2011);
Fuini III et al. J. Phys. G38, 015004 (2011).
Needed very careful tests of convergency
vs. Ntest, Dxcell, # time steps !
Non Isotropic Cross Section - s(q)
Relaxation Time Approximation
hRTA / s =
1
1
< p>
< p > t tr =
15
15 h(a) s TOT r
h(a) = 4a(1+ a)éë(2a +1)ln(1+ a-1 ) - 2ùû , a = mD2 / s
h(a)=str/stot weights cross section by q2
Chapmann-Enskog (CE)
RTA is the one usually emplyed to make theroethical
estimates: Gavin NPA(1985); Kapusta, PRC82(10);
Redlich and Sasaki, PRC79(10), NPA832(10);
Khvorostukhin PRC (2010) …
for a generic cross section:
-2
ds
µ ( q 2 (q ) + mD2 )
dW
mD regulates the angular dependence
Green-Kubo in a box - s(q)
g(a) correct function that fix the
momentum transfer for shear motion
 CE and RTA can differ by about a factor 2-3
 Green-Kubo agrees with CE
S. Plumari et al., PRC86(2012)054902
What is v2?
N(0°) - N(90°)
N(0°) + N(90°)
V2 =
dN/dp
dN/dp
V2 =
dp
Put it very simplistic:
f (pT , f ) = e
v2 (pT , f ) @
pT -d pcos(2f )
T*
dp
4T *
dp
smaller v2
pT
-
N(0°) - N(90°)
N(0°) + N(90°)
If the momentum dp shift is the same
we can expect flatter distribution
are less efficient in build-up v2
pT
What happens at LHC?
Hydro -like
PbPb@2.76 TeV
Full KLN x & p
 At LHC the larger saturation Qs ( ≈ 2.5 GeV) scale makes the effect larger:
- 4ph/s= 2 not sufficient to get close to the data for Th-KLN, but it is sufficient
if one implements both x &p
 Full fKLN implemention change estimate of h/s by about a factor of 3/2
pT-spectra versus h/s(T)
With h/s(T) T2 we cannot reproduce the data:
- RHIC we can recover the spectra by lowering T0by a 30 MeV
- LHC impact of minijets too large one would need a T0 ≈340 MeV ≈ RHIC
Impact of minijets on v2(pT)
Minijets starts to affect v2(pT)
For pT>1.5 GeV
Effect non-negligible
Sensitivity in transport using sameη/s(T)
 Larger sensitivity on h/s (T) at LHC
 Effect larger respect to viscous hydro, but this depends also on df
Viscous Hydrodynamics


T   Tideal
 P dissip
I0 Navier-Stokes, but it violates causality,
II0 order needed -> Israel-Stewart
Teq  d T   f eq  d f
An Asantz (Grad)
h pT2
p  p p
f
df 
f eq »
2 eq
2
3s t T
e P T
K. Dusling et al., PRC81 (2010)
- this implies RTA and not CE
- at pT~3 GeV !? df/f≈ 5
Problems:
 dissipative correction to f -> feq+dfneq just an ansatz
 dfneq/f at pT> 1.5 GeV is large
 dfneq <-> h/s implies a RTA approx. (solvable)
 P (t0) =0 -> discard initial non-eq (ex. minijets)
 pT -> 0 no problem except if h/s is large
Transport at fixed h/s vs Viscous Hydro a test in 3+1D
Changing M of partons one gets
different EoS – cs(T)
Au+Au@200AGeV
v2/ex (0)decrease with cs
 Time scales, trends and value
quite similar to hydro evolution
 An exact comparison under the same
conditions has not been done
stot=15 mb
 r-space: standard Glauber model
 p-space: Boltzmann-Juttner Tmax=1.7-3.5 Tc [pT<2 GeV ]+ minijet [pT>2-3GeV]
We fix maximum initial T at RHIC 200 AGeV
Tmax0 = 340 MeV
T0 0 =1 -> 0=0.6 fm/c
Typical hydro
condition
Then we scale it according to initial e
62 GeV
200 GeV
2.76 TeV
T0
290 MeV
340 MeV
580 MeV
0
0.7 fm/c
0.6 fm/c
0.3 fm/c
Discarded in viscous hydro
 r-space: standard Glauber condition
 p-space: Boltzmann-Juttner Tmax=2(3) Tc [pT<2 GeV ]+ minijet [pT>2-3GeV]
V2 from KLN in Hydro
What does it KLN in hydro?
1) r-space from KLN (larger ex)
2) p-space thermal at t0 ≈0.8 fm/c - No Qs scale , We’ll call it fKLN-Th
Larger ex - > higher h/s to get
the same v2(pT)
Risultati di Heinz da “hadron production………”
Uncertainty on initial conditions
implies uncertainty of a factor 2 on h/s
Heinz et al., PRC 83, 054910 (2011)