1. Early Time Evolution of High Energy Nuclear Collisions Rainer Fries Texas A&M University & RIKEN BNL Early Time Dynamics in Heavy Ion Collisions McGill University, Montreal, July 18, 2007 With J. Kapusta and Y. Li

2. ETD-HIC2 Rainer Fries Motivation ETD Questions ETD Ideas QGP Hydro clQCD CGC pQCD How much kinetic energy is lost in the collision of two nuclei with a total kinetic energy of 40 TeV? How long does it take to decelerate them? How is this energy stored initially? Does it turn into a thermalized plasma? How and when would that happen? Pheno. Models

3. ETD-HIC3 Rainer Fries Motivation Assume 3 overlapping phases:  Initial interaction: energy deposited between the nuclei; gluon saturation, classical fields (clQCD), color glass  Pre-equilibrium / Glasma: decoherence? thermalization? particle production? instabilities?  Equilibrium (?): (ideal ?) hydrodynamics What can we say about the global evolution of the system up to the point of equilibrium? Hydro Non-abelian dynamics clQCD

4. ETD-HIC4 Rainer Fries Outline Goal: space-time map of a high energy nucleus-nucleus collision. Small time expansion of YM; McLerran-Venugopalan model Energy density, momentum, flow Matching to Hydrodynamics Baryon Stopping

5. ETD-HIC5 Rainer Fries Hydro + Initial Conditions Hydro evolution of the plasma from initial conditions  Energy momentum tensor for ideal hydro  + viscous corrections ?  e, p, v, (n B, …) have initial values at  =  0 Goal: measure EoS, viscosities, …  Initial conditions = additional parameters Constrain initial conditions:  Hard scatterings, minijets (parton cascades)  String or Regge based models; e.g. NeXus [Kodama et al.]  Color glass condensate [Hirano, Nara]

6. ETD-HIC6 Rainer Fries Hydro + Initial Conditions Hydro evolution of the plasma from initial conditions  Energy momentum tensor for ideal hydro  + viscous corrections ?  e, p, v, (n B, …) have initial values at  =  0 Assume plasma at  0 created through decay of gluon field F  with energy momentum tensor T f .  Even w/o detailed knowledge of non-abelian dynamics: constraints from energy & momentum conservation for T pl   T f  ! Need gluon field F  and T f  at small times.  Estimate using classical Yang-Mills theory

7. ETD-HIC7 Rainer Fries Classical Color Capacitor 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 classical Yang Mills equation McLerran-Venugopalan model:  For an observable O: average over charge distributions   Gaussian weight [McLerran, Venugopalan]

8. ETD-HIC8 Rainer Fries Color Glass: Two Nuclei Gauge potential (light cone gauge):  In sectors 1 and 2 single nucleus solutions A i 1, A 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 [McLerran, Venugopalan] [Kovner, McLerran, Weigert] [Jalilian-Marian, Kovner, McLerran, Weigert]

9. ETD-HIC9 Rainer Fries Small  Expansion In the forward light cone:  Perturbative solutions [Kovner, McLerran, Weigert]  Numerical solutions [Venugopalan et al; Lappi] Analytic solution for small times? Solve equations in the forward light cone using expansion in time  :  Get all orders in coupling g and sources  ! YM equations In the forward light cone Infinite set of transverse differential equations

10. ETD-HIC10 Rainer Fries Solution can be found recursively to any order in  ! 0 th order = boundary condititions: All odd orders vanish Even orders: Small  Expansion

11. ETD-HIC11 Rainer Fries Note: order in  coupled to order in the fields. Expanding in powers of the boundary fields :  Leading order terms can be resummed in  This reproduces the perturbative KMW result. Perturbative Result In transverse Fourier space

12. ETD-HIC12 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. Corrections to transverse fields at order  3. Gluon Near Field E0E0 B0B0 ☺ ☺

13. ETD-HIC13 Rainer Fries Gluon Near Field Before the collision: transverse fields in the nuclei  E and B orthogonal

14. ETD-HIC14 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

15. ETD-HIC15 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 Transverse E, B fields start to build up linearly

16. ETD-HIC16 Rainer Fries Gluon Near Field Reminiscent of color capacitor  Longitudinal magnetic field of ~ equal strength Strong initial longitudinal ‘pulse’:  Main contribution to the energy momentum tensor [RJF, Kapusta, Li]; [Lappi]; …  Particle production (Schwinger mechanism) [Kharzeev, Tuchin];... Caveats:  Instability from quantum fluctuations? [Fukushima, Gelis, McLerran]  Corrections from violations of boost invariance?

17. ETD-HIC17 Rainer Fries Energy Momentum Tensor Compute energy momentum tensor T f . Initial value of the energy density: Only diagonal contributions at order  0 :  Longitudinal vacuum field Negative longitudinal pressure  maximal anisotropy transv.  long.  Leads to the deceleration of the nuclei Positive transverse pressure  transverse expansion

18. ETD-HIC18 Rainer Fries Energy Momentum Tensor Energy and longitudinal momentum flow at order  1 : Distinguish hydro-like contributions and non-trivial dynamic contributions  Free streaming: flow = –gradient of transverse pressure  Dynamic contribution: additional stress

19. ETD-HIC19 Rainer Fries Energy Momentum Tensor Order O(  2 ): first correction to energy density etc. General structure up to order  3 (rows 1 & 2 shown only) Energy and momentum conservation:

20. ETD-HIC20 Rainer Fries McLerran Venugopalan Model So far just classical YM; add color random walk. E.g. consider initial energy density  0.  Correlator of 4 fields, factorizes into two 2-point correlators:  2-point function G k for nucleus k: Analytic expression for G k in the MV model is known.  Caveat: logarithmically UV divergent for x  0!  Not seen in previous numerical simulations on a lattice.  McLerran-Venugopalan does not describe UV limit correctly; use pQCD [T. Lappi]

21. ETD-HIC21 Rainer Fries Estimating Energy Density Initial energy density in the MV model  Q 0 : UV cutoff   k 2 : charge density in nucleus k from Compatible with estimate using screened abelian boundary fields modulo exact form of logarithmic term. [RJF, Kapusta, Li (2006)]

22. ETD-HIC22 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 Bending around

23. ETD-HIC23 Rainer Fries Transverse Flow @ O(  1 ) Free-streaming part in the MV model. Dynamic contribution vanishes!

24. ETD-HIC24 Rainer Fries Anisotropic Flow Sketch of initial flow in the transverse plane: Clear flow anisotropies for non-central collisions! Caveat: this is flow of energy. b = 8 fm b = 0 fm

25. ETD-HIC25 Rainer Fries Coupling to the Plasma Phase How to get an equilibrated plasma? Use energy-momentum conservation to constrain the plasma phase  Total energy momentum tensor of the system:  r(  ): interpolating function  Enforce

26. ETD-HIC26 Rainer Fries Coupling to the Plasma Phase Here: instantaneous matching  I.e.  Leads to 4 equations to constrain T pl.  Ideal hydro has 5 unknowns: e, p, v Analytic structure of T f  as function of   With etc… Matching to ideal hydro only possible w/o ‘stress’ terms

27. ETD-HIC27 Rainer Fries The Plasma Phase In general: need shear tensor   for the plasma to match. For central collisions (use radial symmetry):  Non-vanishing stress tensor:  Stress indeed related to   p r = radial pressure Need more information to close equations, e.g. equation of state Recover boost invariance y =  (but cut off at  *) Small times:

28. ETD-HIC28 Rainer Fries Application to the MV Model Apply to the MV case  At early times C = 0  Radial flow velocity at early times  Assuming p = 1/3 e  Independent of cutoff

29. ETD-HIC29 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: [Mishustin, Kapusta]

30. ETD-HIC30 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  Rough estimate: [Kapusta, Mishustin] [Mishustin 2006]

31. ETD-HIC31 Rainer Fries Summary Recursive solution for Yang Mills equations (boost- invariant case) Strong initial longitudinal gluon fields Negative longitudinal pressure  baryon stopping Transverse energy flow of energy starts at  = 0 Use full energy momentum tensor to match to hydrodynamics Constraining hydro initial conditions

32. ETD-HIC32 Rainer Fries Backup

33. ETD-HIC33 Rainer Fries Estimating Energy Density Sum over contributions from all charges, recover continuum limit.  Can be done analytically in simple situations  In the following: center of head-on collision of very large nuclei (R A >> R c ) with very slowly varying charge densities  k (x  )   k. E.g. initial energy density  0 :  Depends logarithmically on ratio of scales  = R c Q 0. [RJF, Kapusta, Li]

34. ETD-HIC34 Rainer Fries Energy Matching Total energy content (soft plus pQCD)  RHIC energy.