1. Initial Energy Density, Momentum and Flow in Heavy Ion Collisions Rainer Fries Texas A&M University & RIKEN BNL Heavy Ion Collisions at the LHC: Last Call for Predictions CERN, May 25, 2007

2. LHC: Last Call2 Rainer Fries Outline Space-time map of a high energy nucleus-nucleus collision. Small time expansion in the McLerran-Venugopalan model Energy density, momentum, flow Matching to Hydrodynamics Baryon Stopping In Collaboration with J. Kapusta and Y. Li

3. LHC: Last Call3 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? Initial stage < 1 fm/c Equilibration, hydrodynamics

4. LHC: Last Call4 Rainer Fries Motivation Possible 3 overlapping phases:  Initial interaction: gluon saturation, classical fields (clQCD), color glass  Global evolution of the system + thermalization? particle production? decoherence? instabilities?  Equilibrium, hydrodynamics What can we say about the global evolution of the system up to the point of equilibrium? Hydro Non-abelian dynamics clQCD

5. LHC: Last Call5 Rainer Fries Hydro + Initial Conditions (Ideal?) 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 enter as 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. LHC: Last Call6 Rainer Fries A Simple Model Goal: estimate spatial distribution of energy and momentum at some early time    0.  (Ideal) 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 .  Framework as general as possible w/o details of the dynamics  Constrain T pl  through T f  using energy momentum conservation Use McLerran-Venugopalan model to compute F  and T f  Color Charges J  Class. Gluon Field F  Field Tensor T f  Plasma Tensor T pl  Hydro

7. LHC: Last Call7 Rainer Fries The Starting Point: the MV 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 charge distributions   McLerran-Venugopalan: Gaussian weight [McLerran, Venugopalan]

8. LHC: Last Call8 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. LHC: Last Call9 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. LHC: Last Call10 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. LHC: Last Call11 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. LHC: Last Call12 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. LHC: Last Call13 Rainer Fries Gluon Near Field Before the collision: transverse fields in the nuclei  E and B orthogonal

14. LHC: Last Call14 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. LHC: Last Call15 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. LHC: Last Call16 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];... Caveat: there might be structure on top (corrections from non-boost invariance, fluctuations)

17. LHC: Last Call17 Rainer Fries Energy Momentum Tensor Compute energy momentum tensor T f .  Include random walk over charge distributions  E.g. energy density etc. Initial value of the energy density: Only diagonal contributions at order  0. Energy and longitudinal momentum flow at order  1 :

18. LHC: Last Call18 Rainer Fries Energy Momentum Tensor Initial structure:  Longitudinal vacuum field  Negative longitudinal pressure General structure up to order  3 (rows 1 & 2 shown only) Energy and momentum conservation:

19. LHC: Last Call19 Rainer Fries Energy Momentum Tensor General structure up to order  3 Time hierarchy:  O(  0 ): Initial energy density, pressure  O(  1 ): Transverse ‘flow’  O(  2 ): Decreasing energy density, build-up of other components  O(  3 ): …

20. LHC: Last Call20 Rainer Fries Energy Momentum Tensor General structure up to order  3 Distinguish trivial and non-trivial contributions E.g. flow  Free streaming: flow = –gradient of energy density  Dynamic contribution:

21. LHC: Last Call21 Rainer Fries A Closer Look at Coefficients So far just classical YM; add MV source modeling E.g. consider initial energy density  0.  Contains correlators of 4 fields, e.g.  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!  Naturally not seen in any numerical simulation so far. [T. Lappi]

22. LHC: Last Call22 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. LHC: Last Call23 Rainer Fries Estimating 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.  Can do discrete integrals easily Size of the charges is = 1/Q 0  Scale Q 0 = UV cutoff ! area density of charge

24. LHC: Last Call24 Rainer Fries Estimating the Boundary Fields Field of the single nucleus k:  Estimate non-linearities through screening on scale R c ~ 1/Q s  G = field profile for a single charge contains screening Gives finite correlation function  Logarithmic singularity at x = y recovered for Q 0   What about modes with k T > Q 0 ? Use pQCD.

25. LHC: Last Call25 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]

26. LHC: Last Call26 Rainer Fries Estimating Energy Density Here: central collision at RHIC  Using parton distributions to estimate parton area densities . [McLerran, Gyulassy] 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/fm 3 @ 0.1 fm/c Transverse profile of  0

27. LHC: Last Call27 Rainer Fries Transverse Flow Free-streaming part  Pocket formula derived again for large nuclei and slowly varying charge densities  (center) Transverse profile of the flow slope T 0i free /  for central collisions at RHIC:

28. LHC: Last Call28 Rainer Fries Anisotropic Flow 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

29. LHC: Last Call29 Rainer Fries Coupling to the Plasma Phase How to get an equilibrated (?) plasma?  Difficult! Use energy-momentum conservation to constrain the plasma phase  Total energy momentum tensor of the system:  r(  ): interpolating function  Enforce

30. LHC: Last Call30 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 Matching to ideal hydro is only possible w/o ‘shear’ terms  Tensor in this case:

31. LHC: Last Call31 Rainer Fries The Plasma Phase In general: need shear tensor   for the plasma to match. For central collisions (radial symmetry):  Non-vanishing shear tensor:  Shear 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  *

32. LHC: Last Call32 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]

33. LHC: Last Call33 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]

34. LHC: Last Call34 Rainer Fries Summary Near-field in the MV model  Expansion for small times   Recursive solution known  F , T  : first 4 orders explicitly computed Conclusions:  Strong initial longitudinal fields  Transverse energy flow exists naturally and might be important Constraining initial conditions for hydro  Matching to plasma using energy & momentum conservation  Natural emergence of shear contributions  Estimates of energy densities  Deceleration of charges  baryon stopping

35. LHC: Last Call35 Rainer Fries Backup

36. LHC: Last Call36 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

37. LHC: Last Call37 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

38. LHC: Last Call38 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

39. LHC: Last Call39 Rainer Fries Energy Matching Total energy content (soft plus pQCD)  RHIC energy.