peter kolb, november 18, 2003momentum anisotropies1 momentum anisotropies -- probing the detailed...
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Peter Kolb, November 18, 2003Peter Kolb, November 18, 2003 Momentum AnisotropiesMomentum Anisotropies 11
Momentum Anisotropies --Probing the detailed Dynamics
Department of Physics and AstronomyState University of New YorkStony Brook, NY 11794
with support from theAlexander von Humboldt Foundation
Peter F. Kolb
Brookhaven National LaboratoryFlow and QGP Workshop
November 18, 2003
Peter Kolb, November 18, 2003Peter Kolb, November 18, 2003 Momentum AnisotropiesMomentum Anisotropies 22
Non-Central Collisions Feature Broken Symmetry
Characterize azimuthal dependence of the resulting observables by their Fourier expansion
With the given symmetries and the chosen coordinate system:
no sin terms
no odd cos termsfor equal nuclei and z = 0
Peter Kolb, November 18, 2003Peter Kolb, November 18, 2003 Momentum AnisotropiesMomentum Anisotropies 33
Elliptic Flow - A Prominent Observable
Z.W. Lin and C.M. Ko, PRC 65 (2002) 034904 * requires strong rescattering * requires strong rescattering PFK et al., PLB 500 (2001) 232
D. Molnar and M. Gyulassy, NPA 698 (2002) 379
* … at very early stages* … at very early stages
PFK, J. Sollfrank and U.Heinz, PRC 62 (2000) 054909
* is sensitive on the * is sensitive on the nuclear equation of state nuclear equation of state
PFK and U. Heinz, nucl-ex/0204061Teaney, Lauret, Shuryak, nucl-th/0110037
on a timescale < 1 fm/con a timescale < 1 fm/c
heavier particles pick up larger heavier particles pick up larger <<ppTT> in the common flow field> in the common flow field
Greco, Levai, Ko, Molnar, Voloshin, Greco, Levai, Ko, Molnar, Voloshin, Fries, Nonaka, Bass, Müller, SorensenFries, Nonaka, Bass, Müller, Sorensen
* shows a quark counting rule* shows a quark counting rule
indicating a soft transition regionindicating a soft transition region
common partonic flowcommon partonic flow
reaching the hydrodynamic limitreaching the hydrodynamic limit
* shows a strong hydrodynamic* shows a strong hydrodynamic mass effect mass effect
Huovinen, PFK, Heinz, Ruuskanen, Voloshin, PLB 503 (2001) 58
Peter Kolb, November 18, 2003Peter Kolb, November 18, 2003 Momentum AnisotropiesMomentum Anisotropies 44
Elliptic Flow at Large Pt is HUGE !
PH
EN
IX C
olla
bora
tion
, nuc
l-ex
/030
5013
v2 = 0.25 means that there are 3 times as many particles emitted
into the reaction plane than out of the reaction plane!Furthermore the distribution is not elliptic any longer!
PHENIX Collaboration, nucl-ex/0305013
Peter Kolb, November 18, 2003Peter Kolb, November 18, 2003 Momentum AnisotropiesMomentum Anisotropies 55
Higher order term to Counteract ?
develops a minimum in cartesian coordinates at
x=0 for v2 > 0.1
To balance the minimum a v4 > (10v2-1)/34 is
required (this brings the second derivative of
y(x) at x=0 to vanish).
PFK, Phys. Rev. C 68 (2003) 031902(R)
Peter Kolb, November 18, 2003Peter Kolb, November 18, 2003 Momentum AnisotropiesMomentum Anisotropies 66
A quadrupole modulation of several percent can be expected !
It is however not sufficient to balance the peanut shape
Differential Expectations on Higher v’s
PFK, and R. Rapp, PRC 67 (2003) 044903From a recent hydrodynamic calculation to describe collisions at 200 GeV
PFK, Phys. Rev. C 68 (2003) 031902(R)
Peter Kolb, November 18, 2003Peter Kolb, November 18, 2003 Momentum AnisotropiesMomentum Anisotropies 77
Geometric Anisotropy is Rapidly Converted to Momentum Space
coordinate spacecoordinate space
quantify quantify anisotropiesanisotropies
Peter Kolb, November 18, 2003Peter Kolb, November 18, 2003 Momentum AnisotropiesMomentum Anisotropies 88
A short Digression on Fourier Coefficients
Distribution Distribution within the within the reaction planereaction plane
Distribution Distribution out of the out of the reaction planereaction plane
for a delta-function like distributionfor a delta-function like distribution
Peter Kolb, November 18, 2003Peter Kolb, November 18, 2003 Momentum AnisotropiesMomentum Anisotropies 99
Geometric Anisotropy is Rapdily Converted to Momentum Space
momentum spacemomentum spacecoordinate spacecoordinate space
quantify quantify anisotropiesanisotropies
Peter Kolb, November 18, 2003Peter Kolb, November 18, 2003 Momentum AnisotropiesMomentum Anisotropies 1010
Elliptic Flow in the Blast Wave
Huovinen, PFK, Heinz, Ruuskanen, Voloshin, PLB 503 (01) 58
Blast wave parametrization for non-central collisions
Radial rapidity-field with angular modulation:
Freeze-out on azimuthally symmetric hypersurface of temperature T:
Collapse of the radial integration onto shell:
Peter Kolb, November 18, 2003Peter Kolb, November 18, 2003 Momentum AnisotropiesMomentum Anisotropies 1111
Reproduces
momentum -
and mass dependence
of elliptic anisotropy
Tdec = 140 MeV = 0.58f2 = 7.7 %
Elliptic Flow in the Blast Wave
Peter Kolb, November 18, 2003Peter Kolb, November 18, 2003 Momentum AnisotropiesMomentum Anisotropies 1212
Anisotropic Flow in the Blast Wave
Blast wave parametrization for non-central collisions
Radial rapidity-field with angular modulation:
Freeze-out on azimuthally symmetric hypersurface of temperature T:
Collapse of the radial integration onto shell:
Huovinen, PFK, Heinz, Ruuskanen, Voloshin, PLB 503 (01) 58
Peter Kolb, November 18, 2003Peter Kolb, November 18, 2003 Momentum AnisotropiesMomentum Anisotropies 1313
Blast Wave - Influence of f4
negative negative ff44 positive positive ff44
1 % distortion1 % distortion
2 % distortion2 % distortion
Peter Kolb, November 18, 2003Peter Kolb, November 18, 2003 Momentum AnisotropiesMomentum Anisotropies 1414
Blast Wave - Influence of f6
negative negative ff66 positive positive ff66
1 % distortion1 % distortion
2 % distortion2 % distortion
Peter Kolb, November 18, 2003Peter Kolb, November 18, 2003 Momentum AnisotropiesMomentum Anisotropies 1515
Sensitivity on f2 and f4
A purely elliptic flow anisotropyA purely elliptic flow anisotropyproduces a small quadrupoleproduces a small quadrupolemoment after folding with the moment after folding with the thermal distribution…thermal distribution…
… … an additional small quadropolean additional small quadropoledistortion in the flow field yieldsdistortion in the flow field yieldsa large quadrupole moment ina large quadrupole moment inthe particle distributionthe particle distribution
at some fixed transverse momentum (here pat some fixed transverse momentum (here pTT =2 GeV) =2 GeV)
expandexpand
for small anisotropiesfor small anisotropies
Peter Kolb, November 18, 2003Peter Kolb, November 18, 2003 Momentum AnisotropiesMomentum Anisotropies 1616
pure surface emission in the high pT limit (i.e. from extreme jet-quenching)
Anisotropies from a Black Lense
Cooper Frye freeze-out from the lense with weights on the emission angle
Parameterize surface in the transverse plane,i.e. for
with normal vector
Carry the idea of Shuryak and Voloshin to higher levels (higher orders):PRC 66 (2002) 027902 NPA 715 (2003) 379
for even n
PFK, Phys. Rev. C 68 (2003) 031902(R)
Peter Kolb, November 18, 2003Peter Kolb, November 18, 2003 Momentum AnisotropiesMomentum Anisotropies 1717
Limiting Anisotropies versus Centrality
Extremely interesting centrality dependenceThis picture predicts a peanut like distribution as well!
PFK, Phys. Rev. C 68 (2003) 031902(R)
Peter Kolb, November 18, 2003Peter Kolb, November 18, 2003 Momentum AnisotropiesMomentum Anisotropies 1818
One Thought on Coalescence
assume there are no higher order anisotropies on the quark levelassume there are no higher order anisotropies on the quark level
1/41/4 1/31/3
See talk by Art Poskanzer for experimental results!See talk by Art Poskanzer for experimental results!
2 for mesons2 for mesons3 for baryons3 for baryons
quarksquarks
mesonsmesons baryonsbaryons
Peter Kolb, November 18, 2003Peter Kolb, November 18, 2003 Momentum AnisotropiesMomentum Anisotropies 1919
Summary: Why Study Higher Harmonics ?
Get a full picture of the transverse momentum spectrum
Ideally, triple differentially for different centrality classes:
with odd coefficients at
Anisotropies areAnisotropies are
generated early in the evolution
very sensitve on details of the flow fields and dynamics
probe the hottest stagesprobe the hottest stages
will allow very quantitative statementswill allow very quantitative statementsfrom all model calculationsfrom all model calculations
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