Flow in heavy ion collisions
Urs Achim Wiedemann
CERN PH-TH
Latsis-Symposium, 5 June 2013, Zurich
Heavy Ion Experiments
Elliptic Flow:
hallmark of a collective phenomenon
dN
µ [1+ 2v 2 cos(2f )]
df
Compilation
ALICE, PRL 105, 252302 (2010)
sNN (GeV )
Particle production w.r.t. reaction plane
Particle with
momentum p
Consider single inclusive particle
momentum spectrum
b
f (p) º dN E dp
f
æ
ö
px = pT cos f
ç
÷
p=ç
py = pT sin f
÷
ç p = p 2 + m 2 sinhY ÷
è z
ø
T
To characterize azimuthal asymmetry, measure n-th harmonic moment of f(p).
vn º e
i nf
=
ò
dpei n f f (p)
ò
dp f (p)
n-th order flow
event
average
Problem: This expression cannot be used for data analysis, since the
orientation of the reaction plane is not known a priori.
How to measure flow?
• “Dijet” process
• Maximal asymmetry
• NOT correlated to
the reaction plane
• Many 2->2 or 2-> n
processes
• Reduced asymmetry
~1 N
• final state interactions
• asymmetry caused not only
by multiplicity fluctuations
• collective component is
correlated to the reaction plane
• NOT correlated to
the reaction plane
The azimuthal asymmetry of particle production has a collective
and a random component. Disentangling the two requires a
statistical analysis of finite multiplicity fluctuations.
Measuring flow – one procedure
Want to measure particle production as function of angle w.r.t. reaction plane
●
f
v n ( D) = e i n f
D
But reaction plane is unknown ...
Have to measure particle correlations:
●
e
i n ( f1 - f 2 )
D1 ÙD2
= v n ( D1 ) v n ( D2 ) + e
But this requires signals
●
i n ( f1 - f 2 )
1
vn >
N
Improve measurement with higher cumulants:
corr
D1 ÙD2
“Non-flow effects”
~ O(1 N)
Borghini, Dinh, Ollitrault, PRC (2001)
e i n(f1 +f 2 -f 3 -f 4 ) - e i n(f1 -f 3 ) e i n(f 2 -f 4 ) - e i n(f1 -f 4 ) e i n(f 2 -f 3 ) = -v n4 + O(1 N 3 )
1
This requires signals v n >
N3 4
●
Momentum space
v2 @ LHC
Reaction
plane
dN
µ éë1+ 2 v2 ( pT ) cos ( 2f )ùû
df pT dpT
• Signal v 2 » 0.2 implies 2-1 asymmetry of
particles production w.r.t. reaction plane.
• ‘Non-flow’ effect for 2nd order cumulants
N ~ 100 -1000 Þ1
N ~ 0.1 ~ O(v2 ) ??
2nd order cumulants do not characterize
solely collectivity.
1 N 3 4 ~£ 0.03 << v 2
Strong Collectivity !
pT-integrated v2
The appropriate dynamical framework
●
depends on mean free path
(more precisely: depends on applicability of a quasi-particle picture)
lmfp » ¥ Þ no f - dep lmfp » finite
lmfp < Rsystem
lmfp » 0 Þ max f - dep
f
Theory
tools:
System
Free streaming
p+p
Particle cascade
(QCD transport theory)
Dissipative
fluid dynamics
Perfect fluid
dynamics
?? … pA …?? … AA …
??
The limiting case of perfect fluid dynamics
N im = ni um + ni
(n comp.)
(5 comp.)
Equations of motion
¶m N im º 0
(n constraints)
¶mT mn º 0
(4 constraints)
closed by equation of state
p = p(e,n) (1 constraint)
Dynamical input:
- Initial conditions (uncertainty)
- QCD Equation of state (from Lattice QCD)
- Decoupling (uncertainty)
Tc »175 MeV
Wuppertal-Budapest,
arXiv:1005.3508,
arXiv:1007.2580
Viscous fluid dynamics
Characterizes dissipative corrections in gradient expansion
N im = ni um + ni
(4n comp.)
(10 comp.)
To close equation of motion, supplement conservation laws and eos
¶m N im º 0
¶mT mn º 0
(n constraints)
(4 constraints)
p = p(e,n) (1 constraint)
by point-wise validity of 2nd law of thermodynamics
T¶m S m (x) ³ 0
S m = sum + b qm + O ( Ñ2 )
The resulting Israel-Stewart relativistic fluid dynamics depends in
general on
relaxation times
and
transport coefficients.
Elements of fluid dynamic simulations
Initialization
of thermo-dynamic fields, e.g.
AB
æ 1- x AB
ö
einit ( r ) = e (t 0 , r, h = 0) µ ç
N part (b, r) + xN coll (b, r)÷
è 2
ø
Fluid-dynamic evolution:
Gs =
h
h=0
sT
governs dominant
dissipative mode
Pics by B. Schenke
*
æt0 ö
2
÷ expéë-G s k (t 0 - t )ùû
èt ø
d v(t , k) µ ç
initial
h =1/ 4p
Decoupling: e.g. on space-time hypersurface
,
defined by, T(x) = Tfo
possibly followed by hadronic rescattering
dN i
gi
E
=
dp (2p ) 3
f i ( E, x ) =
ò p.ds (x) f ( p.u(x), x)
i
S
1
exp[( E - mi (x)) T(x)] ± 1
final
S(x)
ds (x)
S(x)
Cooper- Frye
freeze-out
Fluid dynamical models of heavy ion collisions
Compilation by J.-Y. Ollitrault: Plenary IC
Fluid dynamic prior to LHC - results
Fluid dynamics
accounts for:
• Centrality
dependence
of elliptic flow
• pt-dependence of
elliptic flow
P. Romatschke arXiv.0902.3663
• Mass dependence of elliptic
flow (all particle species emerge
from common flow field)
• Single inclusive transverse
momentum spectra at pt (< 3 GeV)
In terms of fluid with
minimal shear viscosity
h
s
<< 1
Implications of minimal viscosity
Back of
envelope:
d(t s) 43 h
=
dt
tT
For 1-dim expanding fluid (Bjorken
boost-invariant), entropy density
increases like
Isentropic “perfect liquid applies if
Put in numbers
t ~ 1 fm /c, T ~ 200 MeV
Theory
s
<< 1
Arnold, Moore, Yaffe,
JHEP 11 (2000) 001
Strong coupling limit
of N=4 SYM
Minimal viscosity
implies strongly
coupled plasma.
 Importance of
strong coupling
techniques
h 1
<< 1
tT s
h
Kovtun, Son, Starinets,
hep-th/0309213
h
l º g2N c
1
>
s 4p
Phenomenological implication
Minimal dissipation  Maximal Transparency to Fluctuations
*
æt ö
d v(t , k) µ ç 0 ÷ expéë-G s k 2 (t 0 - t )ùû
èt ø
1
Fluctuations decay
t 1/e (k) =
on time scale,
Gsk 2
t 1/e (k =1fm-1 ) » 10 - 20 fm
t 1/e ( k = (0.5 fm)-1 ) » 2.5- 5 fm
Models of the initial density distributions in AA-collisions show
generically a set of event-by-event EbyE fluctuations
Fig from M.Luzum, arXiv:1107.0592
Can we see how these spatial eccentricities propagate to
asymmetries vn in momentum distributions?
Flow harmonics from particle correlations @ LHC
Flow harmonics measured via particle correlations.
Here: look directly at correlations of a ‘trigger’ with an ‘associate’ particle
If flow dominated, then
¥
2p dN pairs
=1+ å 2 vn(trig)vn(assoc) cos ( n Df )
N pairs dDf
n=1
Characteristic features:
1. Small-angle jet-like correlations around
Df » Dh » 0 (this is a ‘non-flow’ effect)
2. Long-range rapidity correlation
(almost rapidity-independent ‘flow’)
3. Elliptic flow v2 seems to dominate
(for the semi-peripheral collisions shown here)
4. Away-side peak at Df » p is smaller
(implies non-vanishing odd harmonics v1, v3, …)
ATLAS
prelim
Odd harmonics dominate central collisions
In the most central 0-5% events,
v3 ³ v2
Fluctuations in initial conditions
dominate flow measurements
Flow as linear response to spatial asymmetries
Characterize spatial eccentricities, e.g., via moments of transverse density
e m,n e
infm,n
º-
m inf
r
{ e }
{r }
m
, en º en,n
d x r ( x )...
ò
{...} º
ò d x r ( x)
2
2
ALICE, arXiv:1105.3865, PRL
LHC data indicate:
Spatial eccentricity
is related approx. linearly to
(momentum) flow
v n µ en
Hydrodynamics propagates EbyE fluctuations
• Fluid dynamics maps initial spatial eccentricities onto measured vn
• 3+1 D viscous hydrodynamics
with suitably chosen initial conditions
reproduces v2,v3,v4,v5 in pT and centrality
B. Schenke, MUSIC, .QM2012
Do smaller systems show flow: pPb?
A fluid dynamical simulation of
pPb@LHC yields P. Bozek, 1112.0915
Fluid dynamics compares
surprisingly well with
v2 {2}, v2 {4}, v3 {2}
in pPb@LHC.
ATLAS, 1303.2084
CMS, 1305.0609
A (valid) analogy
From a signal … via fluctuations ….
…. to properties of matter
Slide adapted from W. Zajc
How can non-abelian plasmas thermalize quickly?
• Model-dependent in QCD but a
rigorously calculable problem of
numerical gravity in AdS/CFT
• Very fast non-perturbative
isotropization
t iso <
1
T
M. Heller et al, PRL,
1202.0981
• The first rigorous field theoretic
set-up in which fluid dynamics
applies at very short time scales
as >>1 Þ 0.65£t 0 T0
Chesler, Yaffe,
PRL 102 (2009) 211601
• These non-abelian plasma are unique in that they do not carry
quasi-particle excitations:
perturbatively require
but
t quasi »
const h
T s
t quasi
1
1
~ 2 >>
as T
T
To sum up
• Flow measurements provide an abundant and generic
manifestation of collective dynamics in heavy ion
collisions.
• Fluctuation analyses are still at the beginning.
Directions currently explored include:
system size dependence, event-shape engineering,
mode-by-mode hydrodynamics
• My apologies for not attempting to cover or connect
important other developments in the field of relativistic
heavy ion physics
(jet quenching, quarkonia physics,
thermal photon spectra, open heavy flavor, …, LPV)
End