Early Time Evolution of High Energy Nuclear Collisions

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

2. Motivation 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? • ETD Questions • ETD Ideas How is this energy stored initially? Does it turn into a thermalized plasma? How and when would that happen? QGP Hydro clQCD CGC Pheno. Models pQCD ETD-HIC

3. 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? Non-abelian dynamics clQCD Hydro ETD-HIC

4. 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 ETD-HIC

5. Hydro + Initial Conditions • Hydro evolution of the plasma from initial conditions • Energy momentum tensor for ideal hydro • + viscous corrections ? • e, p, v, (nB, …) 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] ETD-HIC

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

7. 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] ETD-HIC

8. Color Glass: Two Nuclei • Gauge potential (light cone gauge): • In sectors 1 and 2 single nucleus solutions Ai1, Ai2. • 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] ETD-HIC

9. 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 ETD-HIC

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

11. Perturbative Result • 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. In transverse Fourier space ETD-HIC

12. 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. • Corrections to transverse fields at order 3. E0 B0 ☺ ☺ ETD-HIC

13. Gluon Near Field • Before the collision: transverse fields in the nuclei • E and B orthogonal ETD-HIC

14. 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 ETD-HIC

15. 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 • TransverseE, B fields start to build up linearly ETD-HIC

16. 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? ETD-HIC

17. Energy Momentum Tensor • Compute energy momentum tensor Tf. • 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 ETD-HIC

18. 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 ETD-HIC

19. 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: ETD-HIC

20. 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 Gk for nucleus k: • Analytic expression for Gk 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] ETD-HIC

21. Estimating Energy Density • Initial energy density in the MV model • Q0: UV cutoff • k2: charge density in nucleus k from • Compatible with estimate using screened abelian boundary fields modulo exact form of logarithmic term. [RJF, Kapusta, Li (2006)] ETD-HIC

22. 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 ETD-HIC Bending around

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

24. 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 ETD-HIC

25. 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 ETD-HIC

26. Coupling to the Plasma Phase • Here: instantaneous matching • I.e. • Leads to 4 equations to constrain Tpl. • Ideal hydro has 5 unknowns: e, p, v • Analytic structure of Tf as function of  • With etc… • Matching to ideal hydro only possible w/o ‘stress’ terms ETD-HIC

27. 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  • pr = radial pressure • Need more information to close equations, e.g. equation of state Small times: Recover boost invariance y =  (but cut off at *) ETD-HIC

28. 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 ETD-HIC

29. Space-Time Picture [Mishustin, Kapusta] • Finally: field has decayed into plasma at  = 0 • Energy is taken from deceleration of the nuclei in the color field. • Full energy momentum conservation: ETD-HIC

30. 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 • Rough estimate: [Kapusta, Mishustin] [Mishustin 2006] ETD-HIC

31. 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 ETD-HIC

32. Backup ETD-HIC

33. 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 (RA >> Rc) with very slowly varying charge densities k (x)  k. • E.g. initial energy density 0: • Depends logarithmically on ratio of scales  = RcQ0. [RJF, Kapusta, Li] ETD-HIC

34. Energy Matching • Total energy content (soft plus pQCD) • RHIC energy. ETD-HIC