Peter Kolb, November 18, 2003Momentum Anisotropies1 Momentum Anisotropies -- Probing the detailed Dynamics Department of Physics and Astronomy State University of New York Stony Brook, NY with support from the Alexander von Humboldt Foundation Peter F. Kolb Brookhaven National Laboratory Flow and QGP Workshop November 18, 2003

Peter Kolb, November 18, 2003Momentum Anisotropies2 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 terms for equal nuclei and z = 0

Peter Kolb, November 18, 2003Momentum Anisotropies3 Elliptic Flow - A Prominent Observable Z.W. Lin and C.M. Ko, PRC 65 (2002) * 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) * is sensitive on the nuclear equation of state PFK and U. Heinz, nucl-ex/ Teaney, Lauret, Shuryak, nucl-th/ on a timescale < 1 fm/c on a timescale < 1 fm/c heavier particles pick up larger in the common flow field Greco, Levai, Ko, Molnar, Voloshin, Fries, Nonaka, Bass, Müller, Sorensen * shows a quark counting rule indicating a soft transition region common partonic flow reaching the hydrodynamic limit * shows a strong hydrodynamic mass effect * shows a strong hydrodynamic mass effect Huovinen, PFK, Heinz, Ruuskanen, Voloshin, PLB 503 (2001) 58

Peter Kolb, November 18, 2003Momentum Anisotropies4 Elliptic Flow at Large Pt is HUGE ! PHENIX Collaboration, nucl-ex/ v 2 = 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/

Peter Kolb, November 18, 2003Momentum Anisotropies5 Higher order term to Counteract ? develops a minimum in cartesian coordinates at x =0 for v 2 > 0.1 To balance the minimum a v 4 > (10 v 2 -1)/34 is required (this brings the second derivative of y ( x ) at x =0 to vanish). PFK, Phys. Rev. C 68 (2003) (R)

Peter Kolb, November 18, 2003Momentum Anisotropies6 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) From a recent hydrodynamic calculation to describe collisions at 200 GeV PFK, Phys. Rev. C 68 (2003) (R)

Peter Kolb, November 18, 2003Momentum Anisotropies7 Geometric Anisotropy is Rapidly Converted to Momentum Space coordinate space quantify anisotropies

Peter Kolb, November 18, 2003Momentum Anisotropies8 A short Digression on Fourier Coefficients Distribution within the reaction plane Distribution out of the reaction plane for a delta-function like distribution

Peter Kolb, November 18, 2003Momentum Anisotropies9 Geometric Anisotropy is Rapdily Converted to Momentum Space momentum space coordinate space quantify anisotropies

Peter Kolb, November 18, 2003Momentum Anisotropies10 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, 2003Momentum Anisotropies11 Reproduces momentum - and mass dependence of elliptic anisotropy T dec = 140 MeV   = 0.58 f 2 = 7.7 % Elliptic Flow in the Blast Wave

Peter Kolb, November 18, 2003Momentum Anisotropies12 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, 2003Momentum Anisotropies13 Blast Wave - Influence of f 4 negative f 4 positive f 4 1 % distortion 2 % distortion

Peter Kolb, November 18, 2003Momentum Anisotropies14 Blast Wave - Influence of f 6 negative f 6 positive f 6 1 % distortion 2 % distortion

Peter Kolb, November 18, 2003Momentum Anisotropies15 Sensitivity on f 2 and f 4 A purely elliptic flow anisotropy produces a small quadrupole moment after folding with the thermal distribution… … an additional small quadropole distortion in the flow field yields a large quadrupole moment in the particle distribution at some fixed transverse momentum (here p T =2 GeV) expand for small anisotropies

Peter Kolb, November 18, 2003Momentum Anisotropies16 pure surface emission in the high p T 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) NPA 715 (2003) 379 for even n PFK, Phys. Rev. C 68 (2003) (R)

Peter Kolb, November 18, 2003Momentum Anisotropies17 Limiting Anisotropies versus Centrality Extremely interesting centrality dependence This picture predicts a peanut like distribution as well! PFK, Phys. Rev. C 68 (2003) (R)

Peter Kolb, November 18, 2003Momentum Anisotropies18 One Thought on Coalescence assume there are no higher order anisotropies on the quark level 1/41/3 See talk by Art Poskanzer for experimental results! 2 for mesons 3 for baryons quarks mesonsbaryons

Peter Kolb, November 18, 2003Momentum Anisotropies19 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 are generated early in the evolution very sensitve on details of the flow fields and dynamics probe the hottest stages will allow very quantitative statements from all model calculations