1. Elliptic flow from initial states of fast nuclei. A.B. Kaidalov ITEP, Moscow (based on papers with K.Boreskov and O.Kancheli) K.Boreskov and O.Kancheli)

2. Contents: Introduction. Models for elliptic flow. Introduction. Models for elliptic flow. Reggeon diagrams for inclusive cross sections and anisotropic flows. Reggeon diagrams for inclusive cross sections and anisotropic flows. Impact parameter picture. Impact parameter picture. Inclusive vertex. Inclusive vertex. Role of multi-pomeron interactions. Role of multi-pomeron interactions. Conclusions. Conclusions.

3. Introduction. Heavy ions collisions at high energies - a way to study hadronic matter at extreme conditions Т > T c in the deconfinement phase: quark-gluon plasma (QGP). Heavy ions collisions at high energies - a way to study hadronic matter at extreme conditions Т > T c in the deconfinement phase: quark-gluon plasma (QGP).

4. Collisions of heavy ions and QGP. In recent years ideas about properties of QGP have changed (mainly due to RHIC data). In recent years ideas about properties of QGP have changed (mainly due to RHIC data). Previously QGP was usually considered as a Previously QGP was usually considered as a gas of quarks and gluons, while now – as a (nearly ideal) liquid : QGP s. Strong interaction between quarks and gluons is natural at Т ~ 200 MeV ~ Λ QCD. gas of quarks and gluons, while now – as a (nearly ideal) liquid : QGP s. Strong interaction between quarks and gluons is natural at Т ~ 200 MeV ~ Λ QCD. Important role of nonperturbative effects. Important role of nonperturbative effects. Chromomagnetic part of the correlator of gluonic fields does not decrease for T > T c. Chromomagnetic part of the correlator of gluonic fields does not decrease for T > T c.

5. Models for elliptic flow. Anisotropic flows give an important information on role of collective effects in heavy ions collisions. Anisotropic flows give an important information on role of collective effects in heavy ions collisions. Anisotropy of the overlap region leads to anisotropic angular distributions. Anisotropy of the overlap region leads to anisotropic angular distributions. w(φ)=v 0 + ∑ 2v n cos(nφ) v 2 – elliptic flow. v 2 – elliptic flow. v 2 (b,p T,y,s) v 2 (b,p T,y,s) Determined from Determined from event by event event by event analysis analysis

6. Hydrodynamical model. Successful description of RHIC data on V 2 in hydrodynamical model (HM). Successful description of RHIC data on V 2 in hydrodynamical model (HM). Follows from the assumption of local equilibrium. Small mean free path. Follows from the assumption of local equilibrium. Small mean free path. For an ideal fluid (without viscosity) For an ideal fluid (without viscosity) Energy-momentum and baryon number conservation. It is necessary to fix equation Energy-momentum and baryon number conservation. It is necessary to fix equation of state and initial conditions. of state and initial conditions.

7. Elliptic flow and hydro. HM describes dependence of v 2 on р Т ( up to 2 GeV) and on types of particles.

8. Problems of HM. а) Does not describe data for р Т > 2 GeV. Viscosity?

9. Problems of HM. b) Too fast increase with b. b) Too fast increase with b. в) Dependence on y. в) Dependence on y. г) Value of v 4. v 4 / ( v 2 )²= ½; exp: 1.17±0.01 г) Value of v 4. v 4 / ( v 2 )²= ½; exp: 1.17±0.01 Small number of rescatterings and an absence of the local equilibrium. Blaizot et al. Small number of rescatterings and an absence of the local equilibrium. Blaizot et al.

10. Recombination model. Scaling for v 2 /n, p T /n, Scaling for v 2 /n, p T /n, n – number of constituent quarks. n – number of constituent quarks.

11. Final state interaction models of elliptic flow. In the model with final state interaction, which describes suppression of large p T particles v 2 is also reproduced. F.Capella, E.G.Ferreiro F.Capella, E.G.Ferreiro

12. Final state interaction model for elliptic flow. The model reproduces dependence of v 2 on р Т (for all р Т ). Too weak dependence on b.

13. Anisotropy from initial state. Most of theoretical models are based on classical concepts. What is the role of quantum effects? Most of theoretical models are based on classical concepts. What is the role of quantum effects? Field-theoretical approach shows that, Field-theoretical approach shows that, the anisotropy can appear also due to the anisotropy can appear also due to distributions of partons in the initial wave distributions of partons in the initial wave functions of nucleons and nuclei. functions of nucleons and nuclei. К.Boreskov, А.К., О.Kancheli К.Boreskov, А.К., О.Kancheli Thus not only final, but also initial state interactions contribute to v 2 ! Thus not only final, but also initial state interactions contribute to v 2 !

14. Reggeon diagrams for inclusive cross sections and anisotropic flows. Consider single particle inclusive cross sections in the central rapidity region at high energies. For nonenhanced diagrams Consider single particle inclusive cross sections in the central rapidity region at high energies. For nonenhanced diagrams they are expressed in terms of the KM- diagrams they are expressed in terms of the KM- diagrams

15. Reggeon diagrams for inclusive cross sections and anisotropic flows. To find impact parameter representation of the cross section we need the amplitude for nonzero To find impact parameter representation of the cross section we need the amplitude for nonzero

16. Reggeon diagrams for inclusive cross sections and anisotropic flows. where is the nuclear where is the nuclear profile function. profile function. Inclusive vertex Inclusive vertex Odd powers of are absent due to symmetry properties.

17. Elliptic flow. Elliptic flow can be written as Elliptic flow can be written as

18. Estimate of v 2. To estimate the effect use the gaussian parameterization of nuclear form-factors To estimate the effect use the gaussian parameterization of nuclear form-factors

19. Impact parameter picture. Interaction of nuclei in the impact parameter plane Interaction of nuclei in the impact parameter plane

20. Impact parameter picture.

21. In b-space it is easier to understand an origin of the elliptic flow. The inclusive vertex has a corelation between vectors and. Convolution of T int with is sensitive to gradients of nuclear densities. Convolution of T int with is sensitive to gradients of nuclear densities.

22. Inclusive vertex. Feynman diagrams for inclusive vertex Feynman diagrams for inclusive vertex For scalar particles

23. Inclusive vertex. For production of particles with spins For production of particles with spins (e.g. vector mesons) asymmetry is larger and has different sign. Behavior of the vertex at large p t also depends on spins of exchanged particles. In scalar theory In QCD for

24. Two-particle azimuthal correlations. Reggeon diagrams for two-particle inclusive cross sections

25. Two-particle azimuthal correlations. Contribution of Fig. a) has a factorized form Contribution of Fig. a) has a factorized form Diagram of Fig. c) corresponds to non- flow component.

26. Role of multi-pomeron interactions. In the supercritical pomeron theory In the supercritical pomeron theory with role of interactions between pomerons increases with energy and A. with role of interactions between pomerons increases with energy and A.

27. Multi-pomeron interactions. The diagrams of the type b) correspond to interactions between particles from chains produced in different NN-interactions. In this case number of pomerons is the same below and above emission point. These contributions are positive and can lead to a substantial increase of v 2. The diagrams of the type b) correspond to interactions between particles from chains produced in different NN-interactions. In this case number of pomerons is the same below and above emission point. These contributions are positive and can lead to a substantial increase of v 2.

28. Multi-pomeron interactions. Indeed  and Indeed  and R² A  R² A /n. Thus v 2 ~ ². R² A  R² A /n. Thus v 2 ~ ². For central collisions at RHIC energies = 2÷3 and it decreases to =1 for peripheral For central collisions at RHIC energies = 2÷3 and it decreases to =1 for peripheral collisions. collisions.

29. Estimate of v 2. V 2 as a function of p t for two centralities (b=5.6 fm and 7.9 fm) V 2 as a function of p t for two centralities (b=5.6 fm and 7.9 fm)

30. Conclusions. Investigation of anisotropic flows gives an important information on dynamics of collisions and collective effects. Investigation of anisotropic flows gives an important information on dynamics of collisions and collective effects. Elliptic flow is generated not only in Elliptic flow is generated not only in the final state, but also in the initial state interactions. the final state, but also in the initial state interactions. Distributions of partons in colliding nuclei are asymmetric. Distributions of partons in colliding nuclei are asymmetric.