1. Quantum Opacity, RHIC HBT Puzzle, and the Chiral Phase Transition RHIC Physics, HBT and RHIC HBT Puzzle Quantum mech. treatment of optical potential, U (Chiral symmetry), DWEF Reproducing π data Summary, future plans Phys.Rev.Lett.94:102302,2005 and J.Phys.G34:703-740,2007 Gerald Miller and John Cramer, UW

2. The RHIC HBT Puzzle Pratt’s talk- can’t fit entropy and HBT radii with same model Hydrodynamics works BUT NOT FOR HBT

3. q out q side q long R side R long R out p1p1 p2p2 p2p2 + p2p2 p1p1 q Quantum mechanical interference-space time separation of source q=p 1 -p 2 K=(p 1 +p 2 )/2 C(q,K)   p 1,p 2 )   p 1  p 2 ))-1 ~ λ(1-q 2 L R 2 L -q 2 S R 2 S –q 2 O R 2 O ) HBT- 2 particle interferometry Hydrodynamics predicts big R O /R S, Data R O /R S about 1 HBT puzzle

4. Old Formalism  source current density =J  Chaotic sources, Shuryak ‘74 S 0 ~ σ(p 1 )

5. Source Properties “Hydro- Inspired” Emission Function (Bose-Einstein thermal function) (medium density) (Space-time function) includes radial flow

6. Formalism Pions interact U with dense medium  is distorted (not plane) wave Gyulassy et al ‘79 DWEF- distorted wave emission function U -  self energy U :phenomenological-not from equil. thermo, J

7. Wave Equation Solutions Matter is infinitely long Bjorken tube and azimuthal symmetry, wave functions factorize: 3D  2D(distorted)  1D(plane) We solve the reduced Klein-Gordon wave equation for  p : U time independent, cylindrical, Partial wave expansion ! ordinary diff eq

8. Meaning of U Im (U) : Opacity, Re (U) :Refraction pions lose energy and flux Re(U) must exist. Next: very strong attraction chiral phase transition

9. Chiral Symmetry, Son & Stephanov 2002 v 2, v 2 m 2   approach  near T = T c Both terms of U are negative (attractive) =ω 2 -m 2 π

10. Overview Pions emitted anywhere, any time, not only at freeze- out surface Pions interact with the surroundings during escape. These interactions not included in the source function S- No relation between U and S Quarks, gluons are the dominant source of the pions, but not the cause of U Im [U] accounts for opacity Re[U] must exist, causes refraction, acts as mass-change of pions due to chiral-symmetry breaking as they pass from the hot, dense collision medium [m(  )  0]) to the outside vacuum [m(  )  140 MeV]. Relativistic quantum mechanics, solve pion wave equation with partial wave expansion.

11. Time-Independence, Resonances, and Freeze-Out Use of a time-independent phenomenological optical potential does not invoke the mean field approximation and represents an average over a duration  The effects of the optical potential disappear as the system decays. The optical potential also includes the effects of resonances, including the heavy ones. Heavy resonances decay into π’s outside of the plasma. We account for this by computing only that part of the spectra that is related to the pions in the HBT correlation function λ parameter accounts for these

12. Recent Corrections 1.We discovered in November a convergence vs. integration step size problem in our calculation of optical model wave functions. This had no effect on the HBT radii, but had a strong effect on the slope of the spectrum. This problem was corrected by changing from Runge-Kutta to Numerov wave function solutions. 2.We discovered in March that the effects of the strong chemical potential was being applied to the spectrum, but not to the HBT radii. This error was corrected. M. Luzum (UW) 3.The net result, after refitting, is that the “ambiguities” mentioned previously are gone, and the emission temperature of the model has dropped from T=193 MeV to T=161 MeV. The need for a very deep and absorptive optical potential remains. 4.Result: The New Improved DWEF Model (DWEF v.2.1).

13. Fit STAR Data 6 source, 3 optical potential parameters Fit central STAR data at  s NN =200 GeV T=160 MeV, μ π =pion mass reproduce R o, R s, R l reproduce dN   dy (both magnitude and shape) 8 momentum values (i.e., 32 data points) Correct spectrum for contribution of resonances decaying outside the target

14. DWEF Fits to STAR 200 GeV Pion HBT Radii K T (MeV/c) R O (fm) R S (fm) K T (MeV/c) R L (fm) R O /R S K T (MeV/c) Temperature is about 160 MeV

15. Components of DWEF Calculations Red Solid - Full DWEF Yellow Dots - Plane wave (U=0, no flow) Green Short Dash - Re( p^2 term) only, no flow Aqua Long Dash - Im(p^2 term) only, no flow Cyan Dot Dash - Re(Const term) only, no flow Blue 2-Dot Dash - Flow only, U=0 Violet 3-Dot Dash - DWEF with no BE correction K T (MeV/c) R O (fm) R S (fm) K T (MeV/c) Spectrum dN  2 /2  M T dM T dy

16. DWEF Fit to STAR 200 GeV Pion Spectrum Note: accurate prediction of spectrum slope involves subtle cancellations among wave functions K T (MeV/c) Spectrum dN  2 /2  M T dM T dy

17. Meaning of the Parameters Temperature: 160 MeV Transverse flow rapidity: 1.2  v max =0.83 c, v av =0.6 c Pion emission between 6.2 fm/c and 11 fm/c  soft EOS. WS radius: 12 fm = R (Au) + 4.4 fm > R @ SPS Re(U): 0.5 + 0.85 p 2   deep well  strong attraction size consistent with chiral phase transition (.49 +1 p 2 ). Im(U): 0.13 p 2  mfp  8 fm @ K T =1 fm -1  strong absorption  high density Pion chemical potential:   = mass(  )

18. Optical Wave Functions [|  | 2  (b)] K T = 250 MeV/c K T = 600 MeV/c K T = 100 MeV/c Eikonal Approx. Observer DWEF

19. Centrality: 200 GeV Au+Au R O (fm) Au+Au Fit Au+Au Predictions R L (fm) Au+Au Fit Au+Au Predictions R S (fm) Au+Au Fit Au+Au Predictions Space-time parameters R WS, a WS,   are scaled by participant number. Emission duration  is constant. Red: Central Collisions... Indigo: Peripheral Collisions K T (MeV/c)

20. Centrality: 200 GeV Cu+Cu Cu+Cu Predictions Space-time parameters R WS, a WS,   are scaled by participant number. Emission duration  is scaled as A 1/3. Red: Central Collisions... Indigo: Peripheral Collisions R O (fm) Au+Au Fit R S (fm) Au+Au Fit R L (fm) Au+Au Fit K T (MeV/c)

21. Low p T Behavior: Ramsauer Resonances in Well K T (MeV/c) Phobos 0-6% K T (MeV/c) R O (fm)R S (fm) R L (fm) Spectrum dN  2 /2  M T dM T dy

22. Summary and Plans Quantum mechanical treatment of opacity and refraction Excellent fits- many parameters Parameters of U consistent with chiral phase transition, but no relation between U and S implemented Other tests needed- lower energy data –John v 2 Matt Luzum, UW