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Gluon Fields at Early Times and Initial Conditions for Hydrodynamics

Gluon Fields at Early Times and Initial Conditions for Hydrodynamics. Rainer Fries University of Minnesota. with Joe Kapusta, Yang Li. 2006 RHIC/AGS Users’ Meeting June 7, 2006. Introduction. Initial phase of a high energy nuclear collision? Interactions between partons.

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Gluon Fields at Early Times and Initial Conditions for Hydrodynamics

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

  2. 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 Gluon Fields at Early Times

  3. 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, (nB, …) to be determined as functions of , x at  = 0 • Goal: measure EoS, viscosities, … • Initial conditions add more parameters Gluon Fields at Early Times

  4. 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) Gluon Fields at Early Times

  5. 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 Gluon Fields at Early Times

  6. 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 Gluon Fields at Early Times

  7. Outline Hydro Minijets Color Charges J Class. Gluon Field F Field Tensor Tf Plasma Tensor Tpl Gluon Fields at Early Times

  8. 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 Gluon Fields at Early Times

  9. Color Glass: Single Nucleus • Gluon field of single nucleus is transverse • F+ = 0 Fi = 0 Fi+ = (x)i(x) Fij = 0 • Transverse field • Field created by charge fluctuations: • Nucleus is overall color neutral. • Charge  takes random walk in SU(3) space. Longitudinal electric field Ez Longitudinal magnetic field Bz Gluon Fields at Early Times

  10. 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 Gluon Fields at Early Times

  11. 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 Gluon Fields at Early Times

  12. Small  Expansion • Idea: solve equations in the forward light cone using expansion in time  : • 0th order in : • All odd orders vanish: • 2nd 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 Gluon Fields at Early Times

  13. 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 Ez Bz Gluon Fields at Early Times

  14. 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() Ez Bz Gluon Fields at Early Times

  15. Input Fields • Use discrete charge distribution and coarse graining • Assume distribution of quarks & gluons at positions bu in the nuclei. • e.g. charge distribution for nucleus 1 • Tk,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: Gluon Fields at Early Times

  16. Coarse Graining & Screening • Coarse graining • Transverse resolution of the gluon field ~ 1/Qs • Gluon modes with k > Qs: hard processes • Use finite transverse size  ~ 1/Qs for R. • Screening: remove infrared singularity with cutoff Rc. • Impose screening by hand • Then • Rc 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. Gluon Fields at Early Times

  17. 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 Gluon Fields at Early Times

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

  19. Energy Density • Color structure of the longitudinal field: • Energy density • SU(3) random walk for the scalar appearing in  : • It’s really fluctuations: energy ~ N1N2 , field ~ N1N2 Gluon Fields at Early Times

  20. Estimating Energy Density • Energy density created in the center of a head-on collision (x = 0) of large nuclei (RA >> Rc) • Only depends on ratio of scales  = Rc/. • Use approx. constant number density of charges 1, 2 (quarks+antiquarks+9/4 gluons) • Numerical value for Qs = 1 GeV, Rc = 1 fm at RHIC:   450 GeV/fm3. • 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 Gluon Fields at Early Times

  21. Going into the Forward Light Cone • Next coefficient in the energy density, order 2 , is negative. •  expansion takes us to   1/Qs • Match small  expansion and large  asymptotic behavior. • Asymptotics: weak fields at large  (Kovner, McLerran and Weigert) GeV/fm3 O(2) Gluon Fields at Early Times

  22. Going into the Forward Light Cone • Compare to the full result • Numerical result by McLerran & Lappi Preliminary GeV/fm3 O(2) Gluon Fields at Early Times

  23. Energy Momentum Tensor • Early time structure of the energy momentum: • Hierarchy of terms: • Energy and momentum conservation: Gluon Fields at Early Times

  24. 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 : Gluon Fields at Early Times

  25. 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* Gluon Fields at Early Times

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

  27. 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 y0 (Kapusta & Mishustin) • Obtain positions * and rapidities y* of the baryons at  = 0 • Eventually: baryon number distribution • Finally: decay into plasma at  = 0 Gluon Fields at Early Times

  28. 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. Gluon Fields at Early Times

  29. Backup Gluon Fields at Early Times

  30. 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 Fi = 0 Fi+ = (x)i(x) Fij = 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)? Gluon Fields at Early Times

  31. 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 • Ng / Nq ~ 4/Nf after short time, close to chemical equilibrium • Once thermalization is reached: hydrodynamic evolution • Energy momentum tensor of the quark gluon plasma Gluon Fields at Early Times

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

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