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The DWEF Model: Refractive Distortions of HBT John G. Cramer (with Gerald A. Miller) University of Washington Seattle, Washington, USA John G. Cramer (with.

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1 The DWEF Model: Refractive Distortions of HBT John G. Cramer (with Gerald A. Miller) University of Washington Seattle, Washington, USA John G. Cramer (with Gerald A. Miller) University of Washington Seattle, Washington, USA Workshop on Particle Correlations & Femtoscopy - 2007 Santa Rosa, CA August 2, 2007

2 WPCF 20072 Since WPCF 2006 … 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 fugacity from the strong pion chemical potential was being applied to the spectrum, but not to the variables for the HBT radii. This error was corrected. 3.The net result, after refitting, is that the “ambiguities” reported last year are gone, and the emission temperature of the model has dropped from T 0 =193 MeV to T 0 =161 MeV. The need for a very deep and absorptive optical potential remains. 4.Result: The New Improved DWEF Model (DWEF v.2.1).

3 August 2, 2007WPCF 20073 Elements of DWEF Approach: (1) The Nuclear Optical Model 1.Divide the pions into “channels” and focus on pions (Channel 1) that participate in the BE correlation (about 60% of the spectrum pions). Omit “halo” and “resonance” pions and those converted to other particles (Channels 2, 3, etc.). 2.Solve the time-independent Klein-Gordon equation for the wave functions of Channel 1 pions, using a complex potential U. Im( U ) accounts for those pions removed from Channel 1. 3.The complex optical potential U does several things: (a) absorbs pions (opacity); (b) deflects pion trajectories (refraction, demagnification); (c) steals kinetic energy from the emerging pions; (d) produces Ramsauer-type resonances in the well, which can modulate apparent source size and emission intensity. In other words, it quantum-mechanically mocks up the effects on pions of passing through the hot dense medium of the fireball.

4 August 2, 2007WPCF 20074 (2) “Hydro-Inspired” Emission Function (Bose-Einstein thermal function) (medium density) (Space-time function)

5 August 2, 2007WPCF 20075 (3) The DWEF Formalism We use the Wigner distribution of the pion source current density matrix S 0 (x,K) (“the emission function”). The pions interact with the dense medium, producing S(x,K), the distorted wave emission function (DWEF): The  s are distorted (not plane) wave solutions of:, where U is the optical potential. Gyulassy et al., ‘79 Note: assumes chaotic pion sources. Correlation function: Distorted Waves

6 August 2, 2007WPCF 20076 (4) Potential Consistent with Chiral Symmetry Restoration Both terms of U are negative (attractive) U(b)=-(w 0 +w 2 p 2 )  (b), w 0 =real, w 2 =complex Son & Stephanov (2002): v 2 and v 2 m 2  (T) approach 0 near T = T c (We note that this is a low-momentum form of the optical potential that becomes suspect above p~1-2 fm -1 or so.)

7 August 2, 2007WPCF 20077 Data fitting has led us to a chemical potential near the pion mass. We therefore set   = 139.6 MeV = m   We note that the emission temperature favored by the fits (T 0 ~162 MeV) is close to estimates of the temperature for chemical freeze-out, but we leave this as a fit variable. Parameters of the DWEF Model Thermal:T 0 (MeV),   (MeV) (fixed at m  ) Space:R WS (fm), a WS (fm) Time:   (MeV/c),  (MeV/c) Flow:  f (#)  Optical Pot.:Re(w 0 ) (fm -2 ), Re(w 2 ) (#), Im(w 2 ) (#) Wave Eqn.:  (fixed, Kisslinger term off) Total number of parameters: 10 (+2) Note that these parameters describe pion emission at chemical freeze-out, not kinematic freeze-out (e.g., as used in the blast-wave model).

8 August 2, 2007WPCF 20078 DWEF Fits to STAR Data We have calculated pion wave functions in a partial wave expansion, applied them to a “hydro-inspired” pion source function, and calculated the HBT radii and spectrum. This DWEF model uses 7 pion source parameters and 3 optical potential parameters, for a total of 10 parameters in the model. The correlation function C near half-maximum (not the 2 nd moment of C) is calculated. We have fitted STAR data at  s NN =200 GeV, simultaneously fitting R o, R s, R l, and dN p /dy (fitting both magnitude and shape) at 8 momentum values (i.e., 32 data points), using a Levenberg- Marquardt fitting algorithm. In the resulting fit, the  2 per data point is ~3.6 and the  2 per degree of freedom is ~4.8. Only statistical (not systematic) errors are used in calculating  2. We remove long-lived “halo” resonance contributions to the spectrum (which are not included in the model) by multiplying the uncorrected spectrum by ½ (the HBT parameter) before fitting, then “un-correcting” the predicted spectrum with  ½.

9 August 2, 2007WPCF 20079 Components of DWEF Calculations Red Solid - Full DWEF Yellow Dots - Plane wave (W=0, no flow) Green Short Dash - Re(W 2 ) only, no flow Aqua Long Dash - Im(W 2 ) only, no flow Cyan Dot Dash - Re(W 0 ) only, no flow Blue 2-Dot Dash - Flow onlu, W=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

10 August 2, 2007WPCF 200710 Optical Wave Functions [|  | 2  (b)] Full Calculation K T = 197 MeV/c K T = 592 MeV/c K T = 25 MeV/c Imaginary Only Eikonal Approx. Observer Wrong!

11 August 2, 2007WPCF 200711 Optical Wave Functions [|  | 2  (b)] K T = 250 MeV/c K T = 600 MeV/c K T = 100 MeV/c Eikonal Approx. Observer DWEF

12 August 2, 2007WPCF 200712 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)

13 August 2, 2007WPCF 200713 DWEF Fit to STAR 200 GeV Pion Spectrum Note: accurate prediction of spectrum slope involves subtle cancellations among wave functions, which puts severe demands on the numerical accuracy of wave function computations. => The Numerov algorithm. K T (MeV/c) Spectrum dN  2 /2  M T dM T dy

14 August 2, 2007WPCF 200714 The Parameters Temperature: 162 MeV  Chemical freezeout at ~ 160 MeV Transverse flow rapidity: 1.215  v max = 0.84 c, v av = 0.58 c Mean expansion time: 9.1 fm/c  system expands at ~ 0.47 c Pions emitted between 7 fm/c and 11 fm/c  soft EOS. WS radius: 11.9 fm = R(Au) + 4.3 fm > R @ SPS WS diffuseness: 1.1 fm (a bit larger than LENP experience) Re(U): 0.49 + 1.19 p 2  very deep well  strong attraction. Im(U): 0.129 p 2  mfp  8 fm @ K T =1 fm -1  strong absorption  high density Pion chemical potential: take   = m  (pions are massless in the well) We have evidence suggesting a CHIRAL PHASE TRANSITION!

15 August 2, 2007WPCF 200715 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)

16 August 2, 2007WPCF 200716 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)

17 August 2, 2007WPCF 200717 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

18 August 2, 2007WPCF 200718 Summary The improved DWEF Model allows good fits to RHIC HBT radii and spectrum data at emission temperatures of about 162 MeV. We obtain excellent DWEF fits to central STAR  s NN =200 GeV data, simultaneously fitting three HBT radii and the p T spectrum, and we can use participant scaling to predict noncentral Au+Au and Cu+Cu with the same optical potential strengths. The fit parameters are reasonable and indicate strong collective flow, significant opacity, and huge attraction suggesting chiral symmetry restoration. They describe pion emission in hot, highly dense matter with a soft pion equation of state.

19 The End A paper describing this work has been published in Phys. Rev. Lett. 94, 102302 (2005); nucl-th/0411031 ; A longer paper is published in J. Phys. G; nucl-th/0507004

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