1. Gluon Fields at Early Times and Initial Conditions for Hydrodynamics Rainer Fries University of Minnesota 2006 RHIC/AGS Users’ Meeting June 7, 2006 with Joe Kapusta, Yang Li

2. Gluon Fields at Early Times2 Rainer Fries Introduction Initial phase of a high energy nuclear collision?  Interactions between partons.  Energy deposited between the nuclei.  Equilibration, entropy production. Plasma at time  > 0.5 … 1 fm/c.  Hydrodynamic evolution Initial stage < 1 fm/c Equilibration, hydrodynamics

3. Gluon Fields at Early Times3 Rainer Fries Introduction Initial phase of a high energy nuclear collision? Plasma at time  > 0.5 …1 fm/c. Path to equilibrium ?? Hydro evolution of the plasma from initial conditions  , p, v, (n B, …) to be determined as functions of , x  at  =  0 Goal: measure EoS, viscosities, …  Initial conditions add more parameters

4. Gluon Fields at Early Times4 Rainer Fries Introduction Initial phase of a high energy nuclear collision? Plasma at time  > 0.5 …1 fm/c. Path to equilibrium ?? Hydro evolution of the plasma from initial conditions Goal: measure EoS, viscosities Constrain initial conditions:  Hard scatterings, minijets (parton cascades)  String based models  NeXus, HIJING  Color glass + hydro (Hirano, Nara)

5. Gluon Fields at Early Times5 Rainer Fries Color Glass Large nuclei at very large energy: color glass state Saturation  Gluon density sets a scale  High density limit of QCD Large number of gluons in the wave function: classical description of the gluon field

6. Gluon Fields at Early Times6 Rainer Fries Color Glass + Phenomenology Results galore from CGC  Kharzeev, Levin, Nardi ; Kovchegov, Tuchin  Krasnitz and Venugopalan, Lappi Our mission:  Try to understand some of the features analytically  Make contact with phenomenology, hydro  Produce numerical estimates Our approach to deal with this very complex system:  Use simple setup: McLerran-Venugopalan Model (for now …)  Ask the right questions: just calculate energy momentum tensor  Use controlled approximations: e.g. small time expansion  If not possible, make reasonable model assumptions

7. Gluon Fields at Early Times7 Rainer Fries Outline Minijets Color Charges J  Class. Gluon Field F  Field Tensor T f  Plasma Tensor T pl  Hydro

8. Gluon Fields at Early Times8 Rainer Fries The McLerran-Venugopalan Model Assume a large nucleus at very high energy:  Lorentz contraction in longitudinal direction L ~ R/   0  No longitudinal length scale in the problem  boost invariance Replace high energy nucleus by infinitely thin sheet of color charge  Current on the light cone  Solve Yang Mills equations

9. Gluon Fields at Early Times9 Rainer Fries Color Glass: Single Nucleus Gluon field of single nucleus is transverse  F +  = 0 F i  = 0 F i+ =  (x  )  i (x  ) F ij = 0  Transverse field Field created by charge fluctuations:  Nucleus is overall color neutral.  Charge  takes random walk in SU(3) space. Longitudinal electric field E z Longitudinal magnetic field B z

10. Gluon Fields at Early Times10 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

11. Gluon Fields at Early Times11 Rainer Fries Small  Expansion Idea: solve equations in the forward light cone using expansion in time  :  We only believe color glass at small times anyway …  Fields and potentials are regular for   0.  Get all orders in g! Solution can be given recursively! YM equations In the forward light cone Infinite set of transverse differential equations

12. Gluon Fields at Early Times12 Rainer Fries Small  Expansion Idea: solve equations in the forward light cone using expansion in time  :  0 th order in  :  All odd orders vanish:  2 nd order  Arbitrary order in  can be written down. Note: order in  coupled to order in the fields. RJF, J. Kapusta and Y. Li, nucl-th/0604054

13. Gluon Fields at Early Times13 Rainer Fries Gluon Near Field Structure of the field strength tensor Longitudinal electric, magnetic fields start with finite values. For   0 : longitudinal fields = color capacitor? Strong longitudinal pulse (re)discovered recently.  Fries, Kapusta and Li, QM 2005; Kharzeev and Tuchin; Lappi and McLerran, hep-ph/0602189 EzEz BzBz

14. Gluon Fields at Early Times14 Rainer Fries Gluon Near Field Structure of the field strength tensor Longitudinal electric, magnetic fields start with finite values. Transverse E & B fields start at order O(  ) EzEz BzBz

15. Gluon Fields at Early Times15 Rainer Fries Input Fields Use discrete charge distribution and coarse graining Assume distribution of quarks & gluons at positions b u in the nuclei.  e.g. charge distribution for nucleus 1  T k,u = SU(3) matrices  R = profile function of a single charge Write field of these charges in nucleus 1 as  G = field profile for a single charge  In a weak field or abelian limit, this would be the exact solution, e.g. for 2-D Coulomb for point charges:

16. Gluon Fields at Early Times16 Rainer Fries Coarse Graining & Screening Coarse graining  Transverse resolution of the gluon field ~ 1/Q s  Gluon modes with k  > Q s : hard processes  Use finite transverse size ~ 1/Q s for R. Screening: remove infrared singularity with cutoff R c.  Impose screening by hand  Then  R c should depend on the density of charges and should in addition be smaller than 1/  QCD.  This screening should be provided self-consistently by the non- linearities in the YM equations.

17. Gluon Fields at Early Times17 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

18. Gluon Fields at Early Times18 Rainer Fries Charge Fluctuations We have to evaluate  Use discretization: finite but large number of integrals over SU(3) Gaussian weight function for SU(N c ) random walk (Jeon & Venugopalan):  N = number of color charges in the cell around b u, calculated from the number of quarks, antiquarks and gluons.

19. Gluon Fields at Early Times19 Rainer Fries Energy Density Color structure of the longitudinal field: Energy density SU(3) random walk for the scalar appearing in  :  It’s really fluctuations: energy ~ N 1 N 2, field ~  N 1 N 2

20. Gluon Fields at Early Times20 Rainer Fries Estimating Energy Density Energy density created in the center of a head-on collision (x  = 0) of large nuclei (R A >> R c )  Only depends on ratio of scales  = R c /.  Use approx. constant number density of charges  1,  2 (quarks+antiquarks+9/4 gluons) Numerical value for Q s = 1 GeV, R c = 1 fm at RHIC:   450 GeV/fm 3.  Remember: this is for   0.  Scheme for charge density: partons in the wave function minus hard processes. RJF, J. Kapusta and Y. Li, nucl-th/0604054

21. Gluon Fields at Early Times21 Rainer Fries Going into the Forward Light Cone Next coefficient in the energy density, order  2, is negative.  expansion takes us to   1/Q s Match small  expansion and large  asymptotic behavior.  Asymptotics: weak fields at large  (Kovner, McLerran and Weigert) GeV/fm 3 O(2 )O(2 )

22. Gluon Fields at Early Times22 Rainer Fries Going into the Forward Light Cone Compare to the full result Numerical result by McLerran & Lappi GeV/fm 3 Preliminary O(2 )O(2 )

23. Gluon Fields at Early Times23 Rainer Fries Energy Momentum Tensor Early time structure of the energy momentum:  Hierarchy of terms:  Energy and momentum conservation:

24. Gluon Fields at Early Times24 Rainer Fries Matching of the E P Tensors Thermalization? Independent of the mechanism: energy and momentum have to be conserved!   = local energy density, p = pressure Interpolate between the field and the plasma phase  E.g. rapid thermalization around  =  0 :

25. Gluon Fields at Early Times25 Rainer Fries The Plasma Phase Matching gives 4 equations for 5 variables Complete set of equations e.g. by applying equation of state  E.g. for p =  /3: Bjorken: y = , but cut off at some value  *

26. Gluon Fields at Early Times26 Rainer Fries Initial Conditions for the QGP Flow starts to build up linearly with time: System starts to flow before thermalization. Preliminary

27. Gluon Fields at Early Times27 Rainer Fries 3D Space-Time Picture Force acting on the light cone charges  Deceleration of the nuclei;  Trajectory for each bin of mass m: start at beam rapidity y 0 (Kapusta & Mishustin)  Obtain positions  * and rapidities y* of the baryons at  =  0  Eventually: baryon number distribution Finally: decay into plasma at  =  0

28. Gluon Fields at Early Times28 Rainer Fries Summary Problem: how to understand the initial energy and momentum tensor of the plasma from early gluon fields. Introduce small time expansion in the MV model. Estimate initial energy density and its decay with time using a model with discrete, screened charges. Calculate the full energy momentum tensor and match to the plasma phase using energy and momentum conservation.

29. Gluon Fields at Early Times29 Rainer Fries Backup

30. Gluon Fields at Early Times30 Rainer Fries Color Glass: Single Nucleus Current for one nucleus:  Current (in + direction):  Transverse distribution of charge:  (x  )  Solve Yang-Mills equations Gluon field of single nucleus is transverse  F +  = 0 F i  = 0 F i+ =  (x  )  i (x  ) F ij = 0 where  No longitudinal electric or magnetic field in the nuclei.  Transverse electric and magnetic fields are orthogonal to each other. But what is the color distribution  (x  )?

31. Gluon Fields at Early Times31 Rainer Fries Thermalization ? Experimental results indicate thermalization of partons at time scales  0 < 1fm/c Strong longitudinal fields: pair production Numerical work by Lappi: Dirac equation in background field  Quark-antiquark pairs produced copiously  N g / N q ~ 4/N f after short time, close to chemical equilibrium Once thermalization is reached: hydrodynamic evolution  Energy momentum tensor of the quark gluon plasma

32. Gluon Fields at Early Times32 Rainer Fries More Flow This can lead to radial flow early in the plasma phase… … and to elliptic flow b = 8 fm