1. 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

2. QM 20062 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

3. QM 20063 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

4. QM 20064 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 ?

5. QM 20065 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

6. QM 20066 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

7. QM 20067 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

8. QM 20068 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

9. QM 20069 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

10. QM 200610 Rainer Fries Gluon Near Field Before the collision: transverse fields in the nuclei  E and B orthogonal

11. QM 200611 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

12. QM 200612 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

13. QM 200613 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, …)

14. QM 200614 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.

15. QM 200615 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

16. QM 200616 Rainer Fries Energy Momentum Tensor General structure up to order  2 :

17. QM 200617 Rainer Fries Energy Momentum Tensor General structure up to order  2 :

18. QM 200618 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

19. QM 200619 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

20. QM 200620 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/0604054

21. QM 200621 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: 130 GeV/fm3 @ 0.1 fm/c Transverse profile of  0 :  Screening effects: deviations from nuclear thickness scaling

22. QM 200622 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:

23. QM 200623 Rainer Fries Anisotropic Flow Initial flow in the transverse plane: Clear flow anisotropies for non-central collisions b = 8 fm b = 0 fm

24. QM 200624 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:

25. QM 200625 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)

26. QM 200626 Rainer Fries Coupling to the Plasma Phase How to relate field phase and plasma phase? Use energy-momentum conservation to match:  Instantaneous matching

27. QM 200627 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  *

28. QM 200628 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?

29. QM 200629 Rainer Fries Backup

30. QM 200630 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

31. QM 200631 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:

32. QM 200632 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.

33. QM 200633 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

34. QM 200634 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

35. QM 200635 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/0604054

36. QM 200636 Rainer Fries Deceleration through Color Fields Compare (in the McLerran-Venugopalan model):  Fries, Kapusta & Li:  f  260 GeV/fm 3 @  = 0  Lappi:  f  130 GeV/fm 3 @  = 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

37. QM 200637 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 transfer @ t=0 (boost invariant) Charge fluctuations + charge transfer with jets (not boost invariant) IIIIII

38. QM 200638 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