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Early Time Evolution of High Energy Heavy Ion Collisions

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. Outline. Motivation: space-time picture of the gluon field at early times Small time expansion in the McLerran-Venugopalan model

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Early Time Evolution of High Energy Heavy Ion Collisions

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  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. 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 2006

  3. 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 2006

  4. 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. pQCD ? Hydro clQCD QM 2006

  5. Plan of Action Soft modes: hydro evolution from initial conditions • e, p, v, (nB) 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 Tf . • Constrain Tpl through Tf using energy momentum conservation • Use McLerran-Venugopalan model to compute F and Tf Hydro Minijets Color Charges J Class. Gluon Field F Field Tensor Tf Plasma Tensor Tpl QM 2006

  6. Color Glass: Two Nuclei • Gauge potential (light cone gauge): • In sectors 1 and 2 single nucleus solutions i1, i2. • 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 2006

  7. 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 2006

  8. Small  Expansion • Solution can be found recursively to any order in ! • 0th order = boundary condititions: • All odd orders vanish • Even orders: • Note: order in  coupled to order in the fields. • Reproduces perturbative result (Kovner, McLerran, Weigert) QM 2006

  9. Gluon Near Field • Field strength order by order: • Longitudinal electric, magnetic fields start with finite values. • TransverseE, B field start at order : • Corrections to longitudinal fields at order 2: Ez Bz QM 2006

  10. Gluon Near Field • Before the collision: transverse fields in the nuclei • E and B orthogonal QM 2006

  11. 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 2006

  12. 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 • TransverseE, B fields start to build up linearly QM 2006

  13. 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 2006

  14. Energy Density • Initial value : • Contains correlators of 4 fields • Can be factorizes into two 2-point correlators (T. Lappi): • 2-point function Gi for each nucleus i: • Analytic expression for Gi in the MV model is known. • Caveat: logarithmically UV divergent for x  0! • Ergo: MV energy density has divergence for   0. QM 2006

  15. Energy Momentum Tensor • Energy/momentum flow at order 1: • In terms of the initial longitudinal fields Ez and Bz. • No new non-abelian contributions • Corrections at order 2: • E.g. for the energy density Non-abelian correction Abelian correction QM 2006

  16. Energy Momentum Tensor • General structure up to order 2: QM 2006

  17. Energy Momentum Tensor • General structure up to order 2: QM 2006

  18. Compare Full Time Evolution • Compare with the time evolution in numerical solutions (T. Lappi) • The analytic solution discussed so far gives: Asymptotic behavior is known (Kovner, McLerran, Weigert) Normalization Curvature T. Lappi Curvature QM 2006

  19. Modeling the Boundary Fields • Use discrete charge distributions • Coarse grained cells at positions bu in the nuclei. • Tk,u = SU(3) charge from Nk,uq quarks and antiquarks and Nk,ug gluons in cell u. • Size of the charges is  = 1/Q0, coarse graining scale Q0 = UV cutoff • Field of the single nucleus k: • Mean-field: linear field + screening on scale Rc = 1/Qs • G = field profile for a single charge contains screening area density of charge QM 2006

  20. 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 (RA >> Rc) with very slowly varying charge densities k (x)  k. • Depends logarithmically on ratio of scales  = Rc/. RJF, J. Kapusta and Y. Li, nucl-th/0604054 QM 2006

  21. Estimates for T • Here: central collision at RHIC • Using parton distributions to estimate parton area densities  . • Cutoff dependence of Qs and 0 • Qs independent of the UV cutoff. • E.g. for Q0 = 2.5 GeV: 0  260 GeV/fm3. • Compare T. Lappi: 130 GeV/fm3 @ 0.1 fm/c • Transverse profile of 0: • Screening effects: deviations from nuclear thickness scaling QM 2006

  22. 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 2006

  23. Anisotropic Flow • Initial flow in the transverse plane: • Clear flow anisotropies for non-central collisions b = 0 fm b = 8 fm QM 2006

  24. 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 2006

  25. Space-Time Picture • Deceleration: obtain positions * and rapidities y* of the baryons at  = 0 • For given initial beam rapidity y0 , mass area density m. • BRAHMS: • dy = 2.0  0.4 • Nucleon: 100 GeV  27 GeV • We conclude: (Kapusta, Mishustin) QM 2006

  26. Coupling to the Plasma Phase • How to relate field phase and plasma phase? • Use energy-momentum conservation to match: • Instantaneous matching QM 2006

  27. 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 2006

  28. 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 2006

  29. Backup QM 2006

  30. 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 2006

  31. Compare Full Time Evolution • Compare with the time evolution in numerical solutions (T. Lappi) • The analytic solution discussed so far gives: Asymptotic behavior is known (Kovner, McLerran, Weigert) Normalization Curvature Interpolation between near field and asymptotic behavior: T. Lappi O(2) GeV/fm3 Curvature QM 2006

  32. 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 A1, A2) • No further non-abelian terms in the energy-momentum tensor before order 2. QM 2006

  33. 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 Rc. Corrections introduce deviations from original color vector Tu QM 2006

  34. Compute Charge Fluctuations • Integrals discretized: • Finite but large number of integrals over SU(3) • Gaussian weight function for SU(Nc) 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 2006

  35. 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 (RA >> Rc) with very slowly varying charge densities k (x)  k. • Depends logarithmically on ratio of scales  = Rc/. RJF, J. Kapusta and Y. Li, nucl-th/0604054 QM 2006

  36. Deceleration through Color Fields • Compare (in the McLerran-Venugopalan model): • Fries, Kapusta & Li: f  260 GeV/fm3 @  = 0 • Lappi: f  130 GeV/fm3 @  = 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 2006

  37. - - - + + + ’2 ’2 ’1 ’1 2 1 1 1 1 2 2 2 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 I II III Charge fluctuations ~ McLerran-Venugopalan model (boost invariant) Charge fluctuations + charge transfer @ t=0 (boost invariant) Charge fluctuations + charge transfer with jets (not boost invariant) QM 2006

  38. Transverse Structure • Solve expansion around  = 0, simple transverse structure • Effective transverse size 1/ of charges,  ~ Q0 • 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 2 Bin in nucleus 1 Tube with field QM 2006

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