Transcript Effect of fluctuations on v4

Hydrodynamics, flow,
and flow fluctuations
Jean-Yves Ollitrault
IPhT-Saclay
Hirschegg 2010: Strongly Interacting Matter under Extreme Conditions
International Workshop XXXVIII on Gross Properties of Nuclei and Nuclear Excitations
January 19, 2010
In collaboration with Clément Gombeaud and Matt Luzum
Outline
• Elliptic flow and v4
• A simple, universal prediction from ideal
hydrodynamics
• Comparison with data from RHIC
• v4 in ideal and viscous hydrodynamics
• Flow fluctuations, their effect on v4
• Conclusion
Gombeaud, JYO, arXiv:0907.4664, Phys. Rev. C81, 014901 (2010)
Luzum, Gombeaud, JYO, in preparation
Initial azimuthal distribution of particles
Random
parton-parton
collisions
occurring on
scales <<
nuclear radius.
No preferred
direction in the
production
process.
Isotropic
azimuthal
distribution
Final azimuthal distribution of particles
Bar length
= number
of particles
in the
direction
= Azimuthal
(φ)
distribution
plotted in
polar
coordinates
φ
(for pions with transverse momentum ~ 2 GeV/c)
Strong
elliptic flow
created by
pressure
gradients in
the overlap
area
Anisotropic flow
Fourier series expansion of the azimuthal distribution:
Using the φ→-φ and φ→φ+π symmetries of overlap area:
dN/dφ=1+2v2cos(2φ)+2v4cos(4φ)+…
v2=<cos(2φ)> (<…> means average value) is elliptic flow
v4=<cos(4φ)> is a (much smaller) « higher harmonic »
higher harmonics v6, etc are 0 within experimental errors.
This talk is about v4
Azimuthal distribution without v4
A small
effect:
Average
value 0.3%,
maximum
value 3%
Should we
care?
The beauty is in the details!
A primer on hydrodynamics
• Ideal gas (weakly-coupled particles) in global thermal
equilibrium. The phase-space distribution is (Boltzmann)
dN/d3pd3x = exp(-E/T)
Isotropic!
• A fluid moving with velocity v is in (local) thermal
equilibrium in its rest frame:
dN/d3pd3x = exp(-(E-p.v)/T)
Not isotropic: Momenta parallel to v preferred
• At RHIC, the fluid velocity depends on φ:
typically v(φ)=v0+2ε cos(2φ)
The simplicity of v4
• Within the approximation that particle momentum p and
fluid velocity v are parallel (valid for large momentum pt
and low freeze-out temperature T)
dN/dφ=exp(2ε pt cos(2φ)/T)
• Expanding to order ε, the cos(2φ) term is
v2=ε pt/T
• Expanding to order ε2, the cos(4φ) term is
v4=½ (v2)2
Hydrodynamics has a universal prediction for v4/(v2)2 !
Should be independent of equation of state, initial
conditions, centrality, particle momentum and rapidity,
particle type
Borghini JYO nucl-th/0506045
PHENIX results for v4
PHENIX data for charged pions
Au-Au collisions at 100+100 GeV
20-60% most central
The ratio is independent of pT, as predicted by hydro.
The ratio is also independent of particle species within errors.
But… the value is significantly larger than 0.5.
Can detailed (ideal or viscous) hydro calculations explain this?
v4 and coalescence
• The pt range where we have data for v4 is the range where quark
coalescence is thought to be important
• Coalescence predicts, with n_q=2 or 3 constituent quarks
(dN/dφ)hadron(pt)=(dN/dφ)qn_q (pt/n_q)
• Our simple hydrodynamical picture
(dN/dφ)=exp(2ε pt cos(2φ)/T)
is stable under quark coalescence
• In particular, v4/(v2)2 is closer to ½ for hadrons than for parent quarks
• Discrepancy data/hydro is worse for the parent quarks: coalescence
does not help here.
v4 in viscous hydrodynamics
Viscosity changes the fluid evolution and distorts the
momentum distribution of particles emitted at freeze-out.
fviscous(p)=e-E/T(1+δf(p))
where
δf=Cχ(p) (pipj/p2-δij/3)∂iuj
and χ(p) depends on the microscopic interactions and C is
a normalization fixed by matching with the fluid Tμν
Most calculations use χ(p)=p2 (quadratic ansatz) but it has
been recently pointed out that other choices are possible
such as χ(p)=p (linear ansatz)
Dusling Moore Teaney arXiv:0909.0754
Results from ideal and viscous hydro
Hydro parameters tuned to
fit spectra and v2 at RHIC
(Luzum and Romatschke)
In particular, Tf=140 MeV
Ideal hydro predicts a flat
ratio as expected
Viscous hydro with a
linear ansatz is also OK
Viscous hydro with (usual)
quadratic ansatz fails
badly at large pt.
M. Luzum, work in progress
Hydro is unable to explain
a ratio larger than 0.5. We
need something more
More data : centrality dependence
Au-Au collision
100+100 GeV
per nucleon
STAR: Yuting Bai,
PhD thesis Utrecht
PHENIX: Roy Lacey,
private communication
Data > hydro
Small discrepancy between STAR and PHENIX data
Estimating experimental errors
v2 and v4 are not measured
directly but inferred from
azimuthal correlations
(more later on this).
There are many sources of
correlations (jets,
resonance decays,…):
this is the « nonflow » error
which we can estimate
(order of magnitude only)
Difference between STAR and PHENIX data compatible with non-flow error
How do we understand the discrepancy with hydrodynamics??
Eccentricity scaling
y
We understand elliptic flow as the
consequence of the almond shape of the
overlap area
x
 y  x
2
 y  x
2
2
It is therefore natural to expect that v2 scales
like the eccentricity ε of the initial density
profile, defined as :
(this is confirmed by numerical hydro
calculations)

2
Eccentricity fluctuations
Depending on where the
participant nucleons are
located within the nucleus
at the time of the collision,
the actual shape of the
overlap area may vary: the
orientation and eccentricity
of the ellipse defined by
participants fluctuates.
Assuming that v2 scales like the eccentricity,
eccentricity fluctuations translate into
v2 (elliptic flow) fluctuations
We need fluctuations to understand v2 results
(see next talk by Raimond Snellings)
Results using various methods
(STAR)
After correcting for fluctuations
and nonflow
JYO Poskanzer Voloshin, PRC 2009
Eccentricity fluctuations in central collisions
Central collisions are azimuthally symmetric, except for fluctuations:
In the most central bin, v2 and v4 are all fluctuations!
Eccentricity fluctuations are to a good approximation Gaussian in the
transverse plane (2-dimensional Gaussian distribution). This implies
<ε4>=2 <ε2>2
The value of v4 for central collisions rather suggests <(v2)4>~3 <(v2)2>2.
It is therefore unlikely that elliptic flow fluctuations are solely due to
fluctuations in the initial eccentricity.
We need ideas.
Conclusions
• The fourth harmonic, v4, of the azimuthal distribution
gives a further, independent indication that the matter
produced at RHIC expands like a relativistic fluid
• v4 is mostly induced by v2 as a second order effect.
• v4 may help us constrain models based on viscous
hydrodynamics, in particular viscous corrections at
freeze-out : standard quadratic ansatz ruled out?
• v4 is a sensitive probe of elliptic flow fluctuations. The
standard model of eccentricity fluctuations fails for
central collisions. We need a better understanding of
fluctuations.
Backup slides
More results from viscous hydro
Glauber initial
conditions and
smaller viscosity
also reproduces
the measured v2.