1. Energy and Number Density Created at RHIC What’s in the PHENIX White Paper, and a little bit more Paul Stankus, ORNL PHENIX Focus, Apr 11 06

2. 2 Energy Density, Take 1 2R ~ 14 fm 2R/  ~.13 fm Rest Frame  = E/V = M/V 0  ~ 0.14 GeV/fm 3 =   Boosted Frame  = E/V =  M/(V 0 /  ) =  0  2  RHIC = 106  ~ 1570 GeV/fm 3 (!!) Just divide energy by volume, in some frame.

3. 3 Energy Density, Take 2 Examine a box with total momentum zero.  = 0      ~ 3150 GeV/fm 3  = ? Very high, but very short-lived!

4. 4 Energy Density, Take 3 Count up energy in produced particles/matter. Define produced as everything at velocities/rapidities intermediate between those of the original incoming nuclei. Two extremes: All particles Bjorken All fluid Landau

5. 5 The Bjorken Picture: Pure Particles Key ideas: Thin radiator Classical trajectories Finite formation time

6. 6 Particles in a thin box with random velocities Release them suddenly, and let them follow classical trajectories without interactions Strong position- momentum correlations!

7. 7 J.D. Bjorken, Phys. Rev. D 27 (1983) 140 “Highly relativistic nucleus-nucleus collisions: The central rapidity region ” Key idea: Use the space- momentum correlation to translate between spatial density dN/dz and momentum density dN/dp Z Thin radiator The diagram is appropriate for any frame near mid-rapidity, not just the A+A CMS frame specifically.

8. 8 x z p T = p X = p p Z = 0 y = 0 E=√m 2 +p T 2  m T x’ z’ p T = p X m T =√m 2 +p T 2 p Z = m T sinh(y) E = √m 2 +p 2 = m T cosh(y)  Z = p Z /E = sinh(y)/cosh(y) y =tanh -1 (  Z ) y   Z for  Z <<1 Useful relations for particles in different Lorentz frames

9. 9 dZdZ Exercise: Count particles in the green box at some time t, add up their energies, and divide by the volume. Particles in the box iff 0<  Z <dZ/t (limit of infinitely thin source) R Valid for material at any rapidity and for any shape in dE T (t)/dy! A plateau in dE T (t)/dy is not required.

10. 10 How low can t go? Two basic limits:  Bjorken For many years this  Bjorken formula was used with a nominal  Form =1.0 fm/c with no real justification, even when it manifestly violated the crossing time limit for validity! 2R/  = 5.3 fm/c for AGS Au+Au, 1.6 fm/c for SPS Pb+Pb.

11. 11 Better formation time estimates Generic quantum mechanics: a particle can’t be considered formed in a frame faster than h bar /E Translation:  Form  1/m T ~ 1/ PHENIX Data: (dE T /d  )/(dN ch /d  ) ~ 0.85 GeV Assuming 2/3 of particles are charged, this implies  Form ~ 0.35 fm/c

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13. 13 Some assumptions we’ve used Transverse energy density dE T /dy only goes down with time. The number density of particles does not go down with time (entropy conservation). We can estimate, or at least bound, thermalization time from other evidence. An unanswered question: What are the initially produced particles? (Bj: “quanta”)

14. 14 Identifying the intial “quanta” Multiplicities in Au+Au at RHIC were lower than initial pQCD predictions. Indicates need for “regularization”. Good candidate is CGC. CGC identifies intial quanta as high-ish p T gluons (~1 GeV), which is consistent with our particle picture.

15. 15 The Landau Picture: Pure Fluid Key ideas: Complete, instant thermalization Fluid evolves according to ideal relativistic fluid dynamics (1+1) Very simple √s dependences for multiplicity and dN/dy (Gaussian)

16. 16 Courtesy of P. Steinberg; see nucl-ex/0405022 Multiplicities Widths

17. 17 Basic Thermodynamics Sudden expansion, fluid fills empty space without loss of energy. dE = 0 PdV > 0 therefore dS > 0 Gradual expansion (equilibrium maintained), fluid loses energy through PdV work. dE = -PdV therefore dS = 0 Isentropic Adiabatic Hot Cool