Early Time Evolution of High Energy Heavy Ion Collisions Rainer Fries Texas A&M University & RIKEN BNL Talk at Quark Matter 2006, Shanghai November 18, 2006

QM Rainer Fries Outline Motivation: space-time picture of the gluon field at early times Small time expansion in the McLerran-Venugopalan model Energy density Flow Matching to Hydrodynamics In Collaboration with J. Kapusta and Y. Li

QM Rainer Fries Motivation RHIC: equilibrated parton matter after 1 fm/c or less.  Hydrodynamic behavior How do we get there?  Pre-equilibrium phase: energy deposited between the nuclei  Rapid thermalization within less than 1 fm/c Initial stage < 1 fm/c Equilibration, hydrodynamics

QM Rainer Fries Motivation RHIC: equilibrated parton matter after 1 fm/c or less.  Hydrodynamic behavior How do we get there?  Pre-equilibrium phase: energy deposited between the nuclei  Rapid thermalization within less than 1 fm/c Initial dynamics: color glass (clQCD) Later: Hydro How to connect color glass and hydrodynamics?  Compute spatial distribution of energy and momentum at some early time    0.  See also talk by T. Hirano. Hydro pQCD clQCD ?

QM Rainer Fries Plan of Action Soft modes: hydro evolution from initial conditions  e, p, v, (n B ) to be determined as functions of , x  at  =  0 Assume plasma at  0 created through decay of classical gluon field F  with energy momentum tensor T f .  Constrain T pl  through T f  using energy momentum conservation Use McLerran-Venugopalan model to compute F  and T f  Minijets Color Charges J  Class. Gluon Field F  Field Tensor T f  Plasma Tensor T pl  Hydro

QM Rainer Fries Color Glass: Two Nuclei Gauge potential (light cone gauge):  In sectors 1 and 2 single nucleus solutions  i 1,  i 2.  In sector 3 (forward light cone): YM in forward direction:  Set of non-linear differential equations  Boundary conditions at  = 0 given by the fields of the single nuclei Kovner, McLerran, Weigert

QM Rainer Fries Small  Expansion In the forward light cone:  Leading order perturbative solution (Kovner, McLerran, Weigert)  Numerical solutions (Krasnitz, Venugopalan, Nara; Lappi) Our idea: solve equations in the forward light cone using expansion in time  :  We only need it at small times anyway …  Fields and potentials are regular for   0.  Get all orders in coupling g and sources  ! Solution can be given recursively! YM equations In the forward light cone Infinite set of transverse differential equations

QM Rainer Fries Solution can be found recursively to any order in  !  0 th order = boundary condititions:  All odd orders vanish  Even orders: Note: order in  coupled to order in the fields. Reproduces perturbative result (Kovner, McLerran, Weigert) Small  Expansion

QM Rainer Fries Field strength order by order: Longitudinal electric, magnetic fields start with finite values. Transverse E, B field start at order  : Corrections to longitudinal fields at order  2 : Gluon Near Field EzEz BzBz

QM Rainer Fries Gluon Near Field Before the collision: transverse fields in the nuclei  E and B orthogonal

QM Rainer Fries Gluon Near Field Before the collision: transverse fields in the nuclei  E and B orthogonal Immediately after overlap: Strong longitudinal electric, magnetic fields at early times

QM Rainer Fries Gluon Near Field Before the collision: transverse fields in the nuclei  E and B orthogonal Immediately after overlap: Strong longitudinal electric and magnetic field at early times Transverse E, B fields start to build up linearly

QM Rainer Fries Gluon Near Field Reminiscent of color capacitor  Longitudinal magnetic field of equal strength Strong longitudinal pulse: recently renewed interest  Topological charge (Venugopalan, Kharzeev; McLerran, Lappi; …)  Main contribution to the energy momentum tensor (Fries, Kapusta, Li)  Particle production (Kharzeev and Tuchin, …)

QM Rainer Fries Energy Density Initial value :  Contains correlators of 4 fields  Can be factorizes into two 2-point correlators (T. Lappi):  2-point function G i for each nucleus i: Analytic expression for G i in the MV model is known.  Caveat: logarithmically UV divergent for x  0! Ergo: MV energy density has divergence for   0.

QM Rainer Fries Energy Momentum Tensor Energy/momentum flow at order  1 :  In terms of the initial longitudinal fields E z and B z.  No new non-abelian contributions Corrections at order  2 :  E.g. for the energy density Abelian correction Non-abelian correction

QM Rainer Fries Energy Momentum Tensor General structure up to order  2 :

QM Rainer Fries Energy Momentum Tensor General structure up to order  2 :

QM Rainer Fries Compare Full Time Evolution Compare with the time evolution in numerical solutions (T. Lappi) The analytic solution discussed so far gives: Normalization Curvature Asymptotic behavior is known (Kovner, McLerran, Weigert) T. Lappi

QM Rainer Fries Modeling the Boundary Fields Use discrete charge distributions  Coarse grained cells at positions b u in the nuclei.  T k,u = SU(3) charge from N k,u q quarks and antiquarks and N k,u g gluons in cell u.  Size of the charges is = 1/Q 0, coarse graining scale Q 0 = UV cutoff Field of the single nucleus k:  Mean-field: linear field + screening on scale R c = 1/Q s  G = field profile for a single charge contains screening area density of charge

QM Rainer Fries Estimating Energy Density Mean-field: just sum over contributions from all cells Summation can be done analytically in simple situations  E.g. center of head-on collision of very large nuclei (R A >> R c ) with very slowly varying charge densities  k (x  )   k.  Depends logarithmically on ratio of scales  = R c /. RJF, J. Kapusta and Y. Li, nucl-th/

QM Rainer Fries Estimates for T  Here: central collision at RHIC  Using parton distributions to estimate parton area densities . Cutoff dependence of Q s and  0  Q s independent of the UV cutoff.  E.g. for Q 0 = 2.5 GeV:  0  260 GeV/fm 3.  Compare T. Lappi: fm/c Transverse profile of  0 :  Screening effects: deviations from nuclear thickness scaling

QM Rainer Fries Transverse Flow For large nucleus and slowly varying charge densities  :  Initial flow of the field proportional to gradient of the source Transverse profile of the flow slope i /  for central collisions at RHIC:

QM Rainer Fries Anisotropic Flow Initial flow in the transverse plane: Clear flow anisotropies for non-central collisions b = 8 fm b = 0 fm

QM Rainer Fries Space-Time Picture Finally: field has decayed into plasma at  =  0 Energy is taken from deceleration of the nuclei in the color field. Full energy momentum conservation:

QM Rainer Fries Space-Time Picture Deceleration: obtain positions  * and rapidities y* of the baryons at  =  0  For given initial beam rapidity y 0, mass area density  m. BRAHMS:  dy = 2.0  0.4  Nucleon: 100 GeV  27 GeV  We conclude: (Kapusta, Mishustin)

QM Rainer Fries Coupling to the Plasma Phase How to relate field phase and plasma phase? Use energy-momentum conservation to match:  Instantaneous matching

QM Rainer Fries The Plasma Phase Matching gives 4 equations for 5 variables Complete with equation of state  E.g. for p =  /3: Bjorken: y = , but cut off at  *

QM Rainer Fries Summary Near-field in the MV model  Expansion for small times   Recursive solution known Fields and energy momentum tensor: first 3 orders  Initially: strong longitudinal fields  Estimates of energy density and flow Relevance to RHIC:  Deceleration of charges  baryon stopping (BRAHMS)  Matching to plasma using energy & momentum conservation Outlook:  Hydro! Soon.  Connection with hard processes: get rid of the UV cutoff, jets in strong color fields?

QM Rainer Fries Backup

QM Rainer Fries The McLerran-Venugopalan Model Assume a large nucleus at very high energy:  Lorentz contraction L ~ R/   0  Boost invariance Replace high energy nucleus by infinitely thin sheet of color charge  Current on the light cone  Solve Yang Mills equation For an observable O: average over all charge distributions   McLerran-Venugopalan: Gaussian weight

QM Rainer Fries Compare Full Time Evolution Compare with the time evolution in numerical solutions (T. Lappi) The analytic solution discussed so far gives: Normalization Curvature Asymptotic behavior is known (Kovner, McLerran, Weigert) GeV/fm 3 O(2 )O(2 ) T. Lappi Interpolation between near field and asymptotic behavior:

QM Rainer Fries Role of Non-linearities To calculate an observable O: Have to average over all possible charge distributions   We follow McLerran-Venugopalan: purely Gaussian weight Resulting simplifications: e.g. 3-point functions vanish Non-linearities:  Boundary term is non-abelian (commutator of A 1, A 2 )  No further non-abelian terms in the energy-momentum tensor before order  2.

QM Rainer Fries Non-Linearities and Screening Hence our model for field of a single nucleus: linearized ansatz, screening effects from non-linearities are modeled by hand. Connection to the full solution: Mean field approximation:  Or in other words:  H depends on the density of charges and the coupling.  This is modeled by our screening with R c. Corrections introduce deviations from original color vector T u

QM Rainer Fries Compute Charge Fluctuations Integrals discretized:  Finite but large number of integrals over SU(3)  Gaussian weight function for SU(N c ) random walk in a single cell u (Jeon, Venugopalan):  Here: Define area density of color charges: For  0 the only integral to evaluate is

QM Rainer Fries Estimating Energy Density Mean-field: just sum over contributions from all cells  E.g. energy density from longitudinal electric field Summation can be done analytically in simple situations  E.g. center of head-on collision of very large nuclei (R A >> R c ) with very slowly varying charge densities  k (x  )   k.  Depends logarithmically on ratio of scales  = R c /. RJF, J. Kapusta and Y. Li, nucl-th/

QM Rainer Fries Deceleration through Color Fields Compare (in the McLerran-Venugopalan model):  Fries, Kapusta & Li:  f  260 GeV/fm  = 0  Lappi:  f  130 GeV/fm  = 0.1 fm/c Shortcomings:  fields from charges on the light cone  no recoil effects  there are ambiguities in the MV model Net-baryon number = good benchmark test

QM Rainer Fries Color Charges and Currents Charges propagating along the light cone, Lorentz contracted to very thin sheets (  currents J  )  Local charge fluctuations appear frozen (  fluc >>  0 )  Charge transfer by hard processes is instantaneous (  hard <<  0 ) Solve classical EOM for gluon field + - 11 22 ’1’1 ’2’2 + - 11 22 22 11 + - 11 22 ’2’2 ’1’1 Charge fluctuations ~ McLerran-Venugopalan model (boost invariant) Charge fluctuations + charge t=0 (boost invariant) Charge fluctuations + charge transfer with jets (not boost invariant) IIIIII

QM Rainer Fries Transverse Structure Solve expansion around  = 0, simple transverse structure  Effective transverse size 1/  of charges,  ~ Q 0  During time , a charge feels only those charges with transverse distance < c   Discretize charge distribution, using grid of size a ~ 1/   Associate effective classical charge with ensemble of partons in each bin  Factorize SU(3) and x  dependence  Solve EOM for two such charges colliding in opposite bins a Bin in nucleus 1 Bin in nucleus 2 Tube with field