Elliptic flow from initial states

of fast nuclei.

AA..B.B. KaidalovKaidalovITEPITEP, , MoscowMoscow

(based on papers with(based on papers with K.Boreskov and O.Kancheli)K.Boreskov and O.Kancheli)

Contents:Contents:

Introduction.Introduction. Models for elliptic flowModels for elliptic flow.. Reggeon diagrams for inclusive cross Reggeon diagrams for inclusive cross

sections and anisotropic flows.sections and anisotropic flows. Impact parameter picture.Impact parameter picture. Inclusive vertexInclusive vertex.. Role of multiRole of multi--pomeron interactionspomeron interactions.. ConclusionsConclusions..

Introduction.Introduction.

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

Collisions of heavy ions andCollisions of heavy ions and QGPQGP..

In recent years ideas about properties of QGP In recent years ideas about properties of QGP have changed have changed ((mainly due tomainly due to RHICRHIC datadata) .) .

Previously QGP was usually considered as aPreviously QGP was usually considered as a gas of quarks and gluons gas of quarks and gluons , , while nowwhile now – – as aas a

(nearly ideal(nearly ideal) ) liquidliquid :: QGPQGPss.. Strong interaction Strong interaction between quarks and gluons is natural atbetween quarks and gluons is natural at Т Т ~~ 200 200 MeVMeV ~~ ΛΛQCD.QCD.

Important role of nonperturbative effects.Important role of nonperturbative effects. Chromomagnetic part of the correlator of Chromomagnetic part of the correlator of

gluonic fieldsgluonic fields does not decrease fordoes not decrease for T > TT > Tc .c .

Models for elliptic flowModels for elliptic flow..

Anisotropic flows give an important Anisotropic flows give an important information on role of collective information on role of collective effects in heavy ions collisions.effects in heavy ions collisions.

Anisotropy of the overlap region leads Anisotropy of the overlap region leads to anisotropic angular distributions. to anisotropic angular distributions.

w(w(φφ)=v)=v0 0 + ∑ 2v+ ∑ 2vnn cos(ncos(nφφ)) vv2 2 – – elliptic flowelliptic flow.. vv22((b,pb,pTT,y,s),y,s) Determined fromDetermined from event by eventevent by event analysisanalysis

Hydrodynamical modelHydrodynamical model..

Successful description of RHIC data on Successful description of RHIC data on VV2 2 in in hydrodynamical model (HM).hydrodynamical model (HM).

Follows from the assumption of local Follows from the assumption of local equilibriumequilibrium. . Small mean free pathSmall mean free path..

For an ideal fluidFor an ideal fluid ( (without viscositywithout viscosity))

Energy-momentum and baryon number Energy-momentum and baryon number conservationconservation. . It is necessary to fix equationIt is necessary to fix equation

of state and initial conditionsof state and initial conditions..

Elliptic flow and hydroElliptic flow and hydro..

HM describes dependence ofHM describes dependence of v v2 2 onon ррТ (Т (up toup to 2 2 GeVGeV) ) and on types of particlesand on types of particles..

Problems of HMProblems of HM..

а) а) Does not describe data forDoes not describe data for ррТ Т >> 2 2 GeVGeV. . ViscosityViscosity??

Problems of HMProblems of HM..

bb) ) Too fast increase withToo fast increase with bb..

в) в) Dependence onDependence on yy..

г) г) Value of Value of vv4. 4. vv44/ (/ (vv22)²= ½; exp: 1.17±0.01)²= ½; exp: 1.17±0.01

Small number of rescatterings and an absence Small number of rescatterings and an absence of the local equilibriumof the local equilibrium. . Blaizot et al.Blaizot et al.

Recombination model.Recombination model.

Scaling forScaling for vv22/n , p/n , pTT/n , /n ,

n – number of constituent quarksn – number of constituent quarks..

Final state interaction models of Final state interaction models of elliptic flow.elliptic flow.

In the model with final state interaction, In the model with final state interaction, whichwhich describes suppression of large pdescribes suppression of large pT T

particlesparticles vv22 is also reproducedis also reproduced. .

F.Capella, E.G.FerreiroF.Capella, E.G.Ferreiro

Final state interaction model for Final state interaction model for elliptic flow.elliptic flow.

The model reproduces dependence ofThe model reproduces dependence of vv22 on on ррТ Т ((for allfor all р рТТ).).

Too weak dependence onToo weak dependence on bb..

Anisotropy from initial state.Anisotropy from initial state.

Most of theoretical models are based on Most of theoretical models are based on classical conceptsclassical concepts. . What is the role of What is the role of quantum effectsquantum effects??

Field-theoretical approach shows thatField-theoretical approach shows that,, the anisotropy can appear also due tothe anisotropy can appear also due to distributionsdistributions of partons in the initial waveof partons in the initial wave functions offunctions of nucleons and nuclei.nucleons and nuclei. К.К.BoreskovBoreskov,, А.К.,А.К., О.О.KancheliKancheli Thus not only final, but also initial state Thus not only final, but also initial state

interactions contribute tointeractions contribute to vv22 !!

Reggeon diagrams for inclusive cross Reggeon diagrams for inclusive cross sections and anisotropic flows.sections and anisotropic flows.

Consider single particle inclusive cross Consider single particle inclusive cross sections in the central rapidity region at sections in the central rapidity region at high energies. For nonenhanced diagramshigh energies. For nonenhanced diagrams

they are expressed in terms of the KM-they are expressed in terms of the KM-diagramsdiagrams

Reggeon diagrams for inclusive cross Reggeon diagrams for inclusive cross sections and anisotropic flows.sections and anisotropic flows.

To find impact parameter representation To find impact parameter representation of the cross section we need the of the cross section we need the amplitude for nonzero amplitude for nonzero q

Reggeon diagrams for inclusive cross Reggeon diagrams for inclusive cross sections and anisotropic flows.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 Odd powers of are absent due to symmetry properties.properties.

Elliptic flowElliptic flow ..

Elliptic flow can be written asElliptic flow can be written as

Estimate of vEstimate of v2.2.

To estimate the effect use the To estimate the effect use the gaussian parameterization of nuclear gaussian parameterization of nuclear form-factorsform-factors

Impact parameter picture.Impact parameter picture.

Interaction of nuclei in the impact Interaction of nuclei in the impact parameter planeparameter plane

Impact parameter picture.Impact parameter picture.

Impact parameter picture.Impact parameter picture.

In b-space it is easier to understand an In b-space it is easier to understand an origin of the elliptic flow.origin of the elliptic flow.

The inclusive vertex has a corelationThe inclusive vertex has a corelation

between vectors and . between vectors and .

Convolution of TConvolution of Tint int with is with is sensitive to gradients of nuclear sensitive to gradients of nuclear densities.densities.

b

Tp

tp

Inclusive vertexInclusive vertex..

Feynman diagrams for inclusive Feynman diagrams for inclusive vertexvertex

For scalar particlesFor scalar particles

Inclusive vertexInclusive vertex..

For production of particles with spinsFor production of particles with spins

(e.g. vector mesons) asymmetry is(e.g. vector mesons) asymmetry is

larger and has different sign.larger and has different sign.

Behavior of the vertex at large pBehavior of the vertex at large pt t alsoalso

depends on spins of exchanged depends on spins of exchanged particles. In scalar theoryparticles. In scalar theory

In QCD forIn QCD for

Two-particle azimuthal correlations.Two-particle azimuthal correlations.

Reggeon diagrams for two-particle Reggeon diagrams for two-particle inclusive cross sectionsinclusive cross sections

Two-particle azimuthal correlations.Two-particle azimuthal correlations.

Contribution of Fig. a) has a factorized Contribution of Fig. a) has a factorized formform

Diagram of Fig. c) corresponds to non-flow Diagram of Fig. c) corresponds to non-flow component. component.

Role of multiRole of multi--pomeron interactionspomeron interactions.. In the supercritical pomeron theoryIn the supercritical pomeron theory

with role of interactions with role of interactions between pomerons increases with between pomerons increases with energy and A.energy and A.

(0) 1P

MultiMulti--pomeron interactionspomeron interactions..

The diagrams of the type b) The diagrams of the type b) correspond to interactions between correspond to interactions between particles from chains produced in particles from chains produced in different NN-interactions. In this case different NN-interactions. In this case number of pomerons is the same number of pomerons is the same below and above emission point. below and above emission point. These contributions are positive and These contributions are positive and can lead to a substantial increase of can lead to a substantial increase of vv2.2.

MultiMulti--pomeron interactionspomeron interactions..

Indeed Indeed and and

R²R²A A R²R²AA/n . Thus /n . Thus vv2 2 ~~ <n>² .<n>² .

For central collisions at RHIC energies For central collisions at RHIC energies <n> = 2÷3 and it decreases to <n> <n> = 2÷3 and it decreases to <n> =1 for peripheral=1 for peripheral

collisions.collisions.

Estimate of vEstimate of v2 2 ..

VV2 2 as a function of pas a function of pt t for two for two centralities (b=5.6 fm and 7.9 fm)centralities (b=5.6 fm and 7.9 fm)

Conclusions.Conclusions.

Investigation of anisotropic flows Investigation of anisotropic flows gives an important information on gives an important information on dynamics of collisions and collective dynamics of collisions and collective effects.effects.

Elliptic flow is generated not only inElliptic flow is generated not only in the final state, but also in the initial the final state, but also in the initial

state interactions. state interactions. Distributions of partons in colliding Distributions of partons in colliding

nuclei are asymmetric.nuclei are asymmetric.