1 Study Of Collective Flow Phenomena In High Energy Nucleus-Nucleus Collisions L.Chkhaidze, T.Djobava, L.Kharkhelauri Institute of High Energy Physics and Informatization of Tbilisi State Institute of High Energy Physics and Informatization of Tbilisi State University (Georgia) University (Georgia) P. Danielewicz (Michigan State University, USA) P. Danielewicz (Michigan State University, USA) E. Kladnitskaya (JINR, Dubna, Russia) E. Kladnitskaya (JINR, Dubna, Russia) Physics at the Future Colliders, Tbilisi, 2005 Physics at the Future Colliders, Tbilisi, 2005

2 INTRODUCTION The most impressive results of high-energy heavy ion research so far are the new collective phenomena discovered in these reactions. Collective processes are well established both experimentally and theoretically, such as different collective flow patterns. A collective flow is a motion characterized by space-momentum correlations of dynamic origin. Two different signatures of the collective flow have been predicted: a) the bounce-off of compressed matter in the reaction plane (a sidewards deflection of the spectator fragments – “bounce-off”, as well as the directed flow of nucleons from the overlap region between the colliding nuclei in the reaction plane ), called the sideward (also often termed directed) flow; b) the squeeze-out of the participant matter out of the reaction plane – the elliptic flow. The reaction plane is the plane defined by the impact parameter b and the beam direction. A strong dependence of these collective effects to the nuclear equation of state (EOS) is predicted. Nuclear collective (the directed and elliptic) flow has already been observed for protons, light nuclei, pions, K mesons and Λ hyperons emitted in nucleus- nucleus collisions at enegies 0.2 ÷ 200 GeV/c/n at Bevalac, GSI-SIS, Dubna, AGS and CERN/SPS Physics at the Future Colliders, Tbilisi, 2005 Physics at the Future Colliders, Tbilisi, 2005

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4 Many different methods were proposed for the study of flow effects in relativistic nuclear collisions, of which the most commonly used is the method of directivity – the transverse mimentum analysis technique developed by P.Danielewicz and G. Odyniec. They have proposed to present the collective effects as the distribution of average transverse momenta projected onto the reaction plane, as a function of rapidity. Transverse momentum is selected to avoid any possible effects due to nuclear transparency and corona effects that would be manifested primarily in the longitudinal momenta. The transverse momentum technique was used to analyse charged particle exclusive data in the central He-C,C-C, C-Ne, C-Cu and C-Ta interactions at energies of 3.4 ÷ 3.7 GeV per nucleon obtained by SKM-GIBS and Propane Bubble chamber Collaborations of JINR. Physics at the Future Colliders, Tbilisi, 2005 Physics at the Future Colliders, Tbilisi, 2005

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6 TRANSVERSE MOMENTUM ANALYSIS METHOD The method of P.Danielewicz and G. Odyniec involves two basic ideas:The method of P.Danielewicz and G. Odyniec involves two basic ideas: 1) to select the rapidity range and rapdtity dependent waiting factors in the center-of-mass system, which provide the reaction plane closest to the real reaction plane and 2) to remove trivial and spurious self-correlations from the projections. The determination of the reaction plane is crucial in an experiment and it is possible only if the multiplicity, M, of the emitted particles is large and the major part of these particles is detected. Physics at the Future Colliders, Tbilisi, 2005 Physics at the Future Colliders, Tbilisi, 2005

7 DanielewiczTaking into account that the definition of the impact parameter b experimentally is not possible, in the transverse momentum analysis method of Danielewicz and Odyniec and Odyniec the vector b is replaced by the vector Q. The reaction plane vector Q in each individual event is defined by the transverse vector: Q j = ∑ n i=1 ω i P Ti (1) where i is a detected particle index and ω i is a weight. In eq. (1) the vector Q is summing over all protons. ω i =1 for y i > y C + δ, ω i = -1 for y i < y C - δ, and ω i =0 y C -δ < y i < y C +δ. The y C is equal to the value of the overall c.m. system. For He-C and C-C interactions we take δ =0.2, for C-Ne, C-Cu and C-Ta interactions we take δ =0.4. These values of δ remove particles from mid rapidity. For the analysis of C-Ne, C-Cu and C-Ta interactions we chose the events with the number of participant protons N part ≤ 6 and for He-C and C-C collisions respectiveley N part ≤ 3 and N part ≤ 4. The analysis of He-C and C-C interactions have been performed in the c.m.system, while C-Ne, C-Cu and C-Ta have been analysed in lab. system. Danielewicz and OdyniecTo remove the autocorrelations, Danielewicz and Odyniec supposed to to project each particle’s transverse momentum P Ti to a reaction plane determined by all other particles: Q j = ∑ n i≠j ω i P Ti (2) Physics at the Future Colliders, Tbilisi, 2005 Physics at the Future Colliders, Tbilisi, 2005

8 The transverse momentum of each particle, (namely, of protons) in the estimated reaction plane is calculated as: P Xj ´ = ∑ n i≠j ω i · (P Ti · P T j ) / | Q | The dependence of the mean transverse momentum in the reaction plane on the rapidity Y is constructed. The average transverse momentum is obtained by averaging over all events in the corresponding intervals of rapidity. The estimated reaction plane differs from the true one, due to the finite number of particles in each event. The component p X in the true reaction plane is systematically larger than the component in the estimated plane: = / where Φ is the angle between the estimated and true planes. The correction factor k corr = 1 / is the subject of a large uncertainty, especially for Danielewicz low multiplicity. Danielewicz proposed the method for the definition of the correction factor. The k corr have been estimated according to: ≈ [ / ] 1/2 where n is the number of particles in events. From the mean transverse momentum distributions one can extract the transverse flow F, i.e., the slope of the momentum distribution at midrapidity (in the intersection point y=0): F=d /dY Physics at the Future Colliders, Tbilisi, 2005

9 DIRECTED FLOW Clear evidence of in-plane (directed) flow effects for protons, Λ hyperonsClear evidence of in-plane (directed) flow effects for protons, Λ hyperons and π ‾ mesons have been obtained and π ‾ mesons have been obtained. In He-C, C-C and C-Ne interactions directed flow of π ‾ mesons is in the same direction as for protons, while in C- Cu and C-Ta collisions pions show antiflow behavior which is the result of the nuclear shadowing effect. From the transverse momentum distributions of protons, Λ hyperons and π ‾ mesons with respect to the reaction plane, the flow F ( the measure of collective transverse momentum transfer in the reaction plane) have been extracted. The directed flow increases with the mass numbers of projectile A P and target A T. Λ’s and protons flow consistenstly in the same direction. However, the magnitude of the Λ flow is smaller than that for protons. This result is consistent with prior Λ flow measurement performed by E895 Collaboration at AGS in Au-Au collisions at energies of 2, 4, and 6 GeV/nucleon and by EOS collaboration at LBL in Ni-Cu collisions at energy of 2 GeV/nucleon. Physics at the Future Colliders, Tbilisi, 2005 Physics at the Future Colliders, Tbilisi, 2005

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13 He-C C-CC-NeC-CuC-Ta N exp N QGSM F Λ (MeV/c) ± 15 (exp) 97 ± 8 (mod) -- F P exp (MeV/c) 95±8115±11123±12143±15178±20 F P mod (MeV/c) 93±6111±8125±8138±11183±12 F π exp (MeV/c) 17±319±324±5-43±6-74±7 F π mod (MeV/c) 18±3 23±5-43±4-72±4 a 2 exp (p) ± ± ± ± ± a 2 mod (p) ± ± ± ± ± 0.004

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15 F SThe comparison of directed flow results from He-C, C-C, C-Ne,C-Cu and C-Ta interactions with flow data for various projectile/target configurations obtained at BEVALAC, GSI-SIS, AGS and CERN/SPS, was made using the scaled flow F S = F/(A 1/3 P +A 1/3 T ). F S demonsrates a common scaling behaviour for flow values from different systems over all available energy region of 0.2 ÷ 200 GeV/nucleon. QGSM predicts the directed and elliptic flow effects for protons and π ‾ mesonsThe Quark-Gluon String Model (QGSM) was used for the comparison with the experimental data. The QGSM predicts the directed and elliptic flow effects for protons and π ‾ mesons and satisfactorily describes the experimental data. Physics at the Future Colliders, Tbilisi, 2005 Physics at the Future Colliders, Tbilisi, 2005

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17 ELLIPTIC FLOW (SQUEEZE-OUT) The azimuthal angular distributions (φ) of protons in He-C, C-C, C-Ne, C-Cu and C-Ta collisions and Λ hyperons in C-C have been studied. The analysis have been performed for He-C and C-C collisions in c.m. system and for C-Ne, C-Cu and C-Ta collisions in lab. system. The angle φ is the angle of the transverse momentum of each particle in an event with respect to the reaction plane (cos φ =p X / p T ). To treat the data in a quantitative way the azimuthal distributions were fitted by a polinomial: dN/d φ = a 0 (1 + a 1 cos φ + a 2 cos 2φ) a 2 The anisotropy factor a 2 is negative for out-of-plane enhancement (squeeze-out) and is the measure of the strength of the anisotropic emission. The azimuthal angular distributions show maxima at φ =90 0 and φ =270 0 with respect to the event plane. These maxima of the distributions are associated with preferential particle emission perpendicular to the reaction plane (squeeze-out). Thus a clear signature of an out-of-plane signal (elliptic flow) is evidenced. Physics at the Future Colliders, Tbilisi, 2005 Physics at the Future Colliders, Tbilisi, 2005

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20 A P A TThe elliptic flow is increasing with increasing the transverse momentum and the mass number of the projectile A P, target A T and also with narrowing of the cut applied around the center of mass rapidity. In experiments E895, E877, at AGS and NA49 at CERN SPS the elliptic flow is typically studied at midrapidity and is quantified in terms of the second Fourier coefficient v 2 = v 2 a 2 The Fourier coefficient v 2 is related to a 2 via v 2 = a 2 /2 We have estimated v 2 for all analysed pairs of nuclei. The excitation function v 2 changes the sign from negative to positive elliptic flow within the region of 2.0 ≤ E beam ≤ 8.0 GeV/nucleon i.e. shows the transition from out of plane enhancement to preferential in- plane emission and point to apparent transition energy E tr ~ 4 GeV/nucleon. Physics at the Future Colliders, Tbilisi, 2005 Physics at the Future Colliders, Tbilisi, 2005

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22 Conclusions The collective flow of protons and π ‾ mesons in He-C, C-C, C-Ne, C- Cu and C-Ta collisions and flow of Λ hyperons in C-C at a momentum of 4.2 ÷ 4.5 GeV/c per nucleon (E=3.4 ÷ 3.7 GeV/nucleon) have been investigated. 1.The transverse momentum technique of P. Danielewicz and G.Odyniec was used for data analysis. Clear evidence of directed (in-plane) flow effects for protons, Λ hyperons and π ‾ mesons has been obtained. From the transverse momentum distributions of protons, Λ hyperons and π ‾ mesons with respect to the reaction plane, the flow F (the measure of the collective transverse momentum transfer in the reaction plane) has been extracted. The dependence of F on mass numbers of projectile A P and target A T was studied. The absolute value of F increases linearly with the increase of A P and A T. 2.A clear signature of elliptic flow (squeeze-out) has been obtained from the azimuthal distributions of protons with respect to the reaction plane in He-C, C-C, C-Ne, C-Cu and C-Ta and Λ hyperons in C-C collisions collisions. Azimuthal distributions have been parametrized by a second order polynomial function, the parameter a 2 of the anisotropy term a 2 cos2 φ have been extracted. The second Fourier coefficient v 2, which is related to a 2 via v 2 =a 2 /2 and measures the elliptic flow, have been estimated both for all pairs of nuclei. The obtained results of the elliptic flow excitation function for protons in Physics at the Future Colliders, Tbilisi, 2005

23 He-C, C-C, C-Ne, C-Cu and C-Ta collisions has been compared with the data in the available energy region of 0.2 ÷ GeV/nucleon. The excitation function v 2 clearly shows an evolution from negative v 2 0 elliptic flow within the region of 2.0 ≤ E beam ≤ 8.0 GeV/nucleon and point to an apparent transition energy E tr ~ 4 GeV/nucleon. 3. All experimental results have been compared with the predictions of the Quark Gluon String Model. The model reproduces experimental data. Physics at the Future Colliders, Tbilisi, 2005