1. STAR Pion Entropy and Phase Space Density at RHIC John G. Cramer Department of Physics University of Washington, Seattle, WA, USA Second Warsaw Meeting on Particle Correlations and Resonances in Heavy Ion Collisions Warsaw University of Technology October 16, 2003 Second Warsaw Meeting on Particle Correlations and Resonances in Heavy Ion Collisions Warsaw University of Technology October 16, 2003

2. STAR October 16, 2003 John G. Cramer2 Phase Space Density: Definition & Expectations Phase Space Density - The phase space density f(p,x) plays a fundamental role in quantum statistical mechanics. The local phase space density is the number of pions occupying the phase space cell at (p,x) with 6-dimensional volume  p 3  x 3 = h 3. The source-averaged phase space density is  f(p)  ∫[f(p,x)] 2 d 3 x / ∫f(p,x) d 3 x, i.e., the local phase space density averaged over the f-weighted source volume. Because of Liouville’s Theorem, for free-streaming particles  f(p)  is a conserved Lorentz scalar. At RHIC, with about the same HBT source size as at the CERN SPS but with more emitted pions, we expect an increase in the pion phase space density over that observed at the SPS.

3. STAR October 16, 2003 John G. Cramer3 hep-ph/0212302 Entropy: Calculation & Expectations Entropy – The pion entropy per particle S  /N  and the total pion entropy at midrapidity dS  /dy can be calculated from  f(p) . The entropy S of a colliding heavy ion system should be produced mainly during the parton phase and should grow only slowly as the system expands and cools. Entropy is conserved during hydrodynamic expansion and free- streaming. Thus, the entropy of the system after freeze-out should be close to the initial entropy and should provide a critical constraint on the early- stage processes of the system. nucl-th/0104023 A quark-gluon plasma has a large number of degrees of freedom. It should generate a relatively large entropy density, up to 12 to 16 times larger than that of a hadronic gas. At RHIC, if a QGP phase grows with centrality we would expect the entropy to grow strongly with increasing centrality and participant number. Can Entropy provide the QGP “Smoking Gun”??

4. STAR October 16, 2003 John G. Cramer4 Pion Phase Space Density at Midrapidity The source-averaged phase space density  f(m T )  is the dimensionless number of pions per 6-dimensional phase space cell h 3, as averaged over the source. At midrapidity  f(m T )  is given by the expression: Momentum Spectrum HBT “momentum volume” V p Pion Purity Correction Jacobian to make it a Lorentz scalar Average phase space density

5. STAR October 16, 2003 John G. Cramer5 Changes in PSD Analysis since QM-2002 At QM-2002 (Nantes) we presented a poster on our preliminary phase space density analysis, which used the 3D histograms of STAR Year 1 HBT analysis from our PRL. At QM-2002 (see Scott Pratt’s summary talk) we also started our investigation of the entropy implications of the PSD. This analysis was also reported at the INT/RHIC Winter Workshop, January – 2003 (Seattle). CHANGES: We have reanalyzed the STAR Year 1 data (S nn ½ = 130 GeV) into 7 centrality bins for |y| < 0.5, incorporating several improvements : 1.We use 6 K T bins (average pair momentum) rather than 3 p T bins (individual pion momentum) for pair correlations (better large-Q statistics). 2.We limit the vertex z-position to ±55 cm and bin the data in 21 z-bins, performing event mixing only between events in the same z-bin. 3.We do event mixing only for events in ±30 0 of the same reaction plane. 4.We combined     and     correlations (improved statistics). 5.We used the Bowler-Sinyukov-CERES procedure and the Sinyukov analytic formula to deal with the Coulomb correction. (We note that Bowler Coulomb procedure has the effect of increasing radii and reducing, thus reducing the PSD and increasing entropy vs. QM02.) We also found and fixed a bug in our PSD analysis program, which had the effect of systematically reducing for the more peripheral centralities. This bug had no effect on the 0-5% centrality.

6. STAR October 16, 2003 John G. Cramer6 RHIC Collisions as Functions of Centrality 50-80%30-50%20-30%10-20%5-10%0-5% At RHIC we can classify collision events by impact parameter, based on charged particle production. Participants Binary Collisions Frequency of Charged Particles produced in RHIC Au+Au Collisions of  Total

7. STAR October 16, 2003 John G. Cramer7 Corrected HBT Momentum Volume V p / ½ STAR Preliminary Central Peripheral m T - m  (GeV) 0-5% 5-10% 10-20% 20-30% 30-40% 40-50% 50-80% Centrality Fits assuming: V p  ½ =A 0 m T 3  (Sinyukov)

8. STAR October 16, 2003 John G. Cramer8 Global Fit to Pion Momentum Spectrum We make a global fit of the uncorrected pion spectrum vs. centrality by: (1)Assuming that the spectrum has the form of an effective- T Bose-Einstein distribution: d 2 N/m T dm T dy=A/[Exp(E/T) –1] and (2)Assuming that A and T have a quadratic dependence on the number of participants N p : A(p) = A 0 +A 1 N p +A 2 N p 2 T(p) = T 0 +T 1 N p +T 2 N p 2 STAR Preliminary

9. STAR October 16, 2003 John G. Cramer9 Interpolated Pion Phase Space Density  f  at S ½ = 130 GeV Central Peripheral NA49 STAR Preliminary Note failure of “universal” PSD between CERN and RHIC. } HBT points with interpolated spectra

10. STAR October 16, 2003 John G. Cramer10 Extrapolated Pion Phase Space Density  f  at S ½ = 130 GeV Central Peripheral STAR Preliminary Spectrum points with extrapolated HBT V p / 1/2 Note that for centralities of 0-40% of  T,  f  changes very little.  f  drops only for the lowest 3 centralities.

11. STAR October 16, 2003 John G. Cramer11 Converting Phase Space Density to Entropy per Particle (1) Starting from quantum statistical mechanics, we define: To perform the space integrals, we assume that f(x,p) =  f(p)  g(x), where g(x) =  2 3 Exp[  x 2 /2R x 2  y 2 /2R y 2  z 2 /2R z 2 ], i.e., that the source has a Gaussian shape based on HBT analysis of the system. Further, we make the Sinyukov-inspired assumption that the three radii have a momentum dependence proportional to m T . Then the space integrals can be performed analytically. This gives the numerator and denominator integrands of the above expression factors of R x R y R z = R eff 3 m T   (For reference,  ~½) An estimate of the average pion entropy per particle  S/N  can be obtained from a 6-dimensional space-momentum integral over the local phase space density f(x,p): O(f) O(f 2 ) O(f 3 )O(f 4 ) f dS 6 (Series)/dS 6 +0.2%  0.2%  0.1%  0.1% 

12. STAR October 16, 2003 John G. Cramer12 Converting Phase Space Density to Entropy per Particle (2) The entropy per particle  S/N  then reduces to a momentum integral of the form: We obtain  from the momentum dependence of V p -1/2 and perform the momentum integrals numerically using momentum-dependent fits to  f  or fits to V p -1/2 and the spectra. (6-D) (3-D) (1-D)

13. STAR October 16, 2003 John G. Cramer13 To integrate over the phase space density, we need a function of p T with some physical plausibility that can put a smooth continuous function through the PSD points. For a static thermal source (no flow), the pion PSD must be a Bose-Einstein distribution: Static = {Exp[(m Total   )/T 0 ]  1}  1. This suggests fitting the PSD with a Bose-Einstein distribution that has been blue-shifted by longitudinal and transverse flow. The form of the local blue-shifted BE distribution is well known. We can substitute for the local longitudinal and transverse flow rapidities  L and  T, the average values and to obtain: Blue-Shifted Bose-Einstein Functions We assume   = =0 and consider three models for : BSBE 1 : =  (i.e., constant average flow, independent of p T ) BSBE 2 : =  (p T /m T ) =  T (i.e., proportional to pair velocity) BSBE 3 : =    T    T 3    T 5    T 7 (minimize  S/N)/  flow)

14. STAR October 16, 2003 John G. Cramer14 Fits to Interpolated Pion Phase Space Density Central Peripheral STAR Preliminary Warning: PSD in the region measured contributes only about 60% to the average entropy per particle. HBT points with interpolated spectra Fitted with BSBE 2 function

15. STAR October 16, 2003 John G. Cramer15 Fits to Extrapolated Pion Phase Space Density Central Peripheral STAR Preliminary Spectrum points with extrapolated HBT V p / 1/2 Each successive centrality reduced by 3/2 Solid = Combined V p / 1/2 and Spectrum fits Dashed = Fitted with BSBE 2 function

16. STAR October 16, 2003 John G. Cramer16 Large-m T behavior of three BSBE Models Solid = BSBE 2 :  T =  T Dotted = BSBE 3 : 7 th order odd polynomial in  T Dashed = BSBE 1 :  T = Constant Each successive centrality reduced by 3/2

17. STAR October 16, 2003 John G. Cramer17 Large m T behavior using Radius & Spectrum Fits Solid = fits to spectrum and V p / 1/2 Dashed = BSBE 2 fits to extrapolated data Each successive centrality reduced by 3/2

18. STAR October 16, 2003 John G. Cramer18 Entropy per Pion from V p / ½ and Spectrum Fits Central Peripheral STAR Preliminary Black = Combined fits to spectrum and V p / 1/2

19. STAR October 16, 2003 John G. Cramer19 Entropy per Pion from BSBE Fits Central Peripheral STAR Preliminary Green = BSBE 2 : ~  T Red = BSBE 1 : Const Blue = BSBE 3 : Odd 7 th order Polynomial in  T Black = Combined fits to spectrum and V p / 1/2

20. STAR October 16, 2003 John G. Cramer20   = 0   = m  Thermal Bose-Einstein Entropy per Particle The thermal estimate of the  entropy per particle can be obtained by integrating a Bose-Einstein distribution over 3D momentum:   /m  T/m  Note that the thermal-model entropy per particle usually decreases with increasing temperature T and chemical potential  .

21. STAR October 16, 2003 John G. Cramer21 Entropy per Particle S/N with Thermal Estimates Central Peripheral STAR Preliminary Dashed line indicates systematic error in extracting V p from HBT. Dot-dash line shows S/N from BDBE 2 fits to  f  Solid line and points show S/N from spectrum and V p / 1/2 fits. For T=110 MeV, S/N implies a pion chemical potential of   =44.4 MeV.

22. STAR October 16, 2003 John G. Cramer22 Total Pion Entropy dS  /dy STAR Preliminary Dashed line indicates systematic error in extracting V p from HBT. Dot-dash line indicates dS/dy from BSBE x fits to interpolated. Solid line is a linear fit through (0,0) with slope = 6.58 entropy units per participant Entropy content of nucleons + antinucleons P&P Why is dS  /dy linear with N p ??

23. STAR October 16, 2003 John G. Cramer23 Initial collision overlap area is roughly proportional to N p 2/3 Initial collision entropy is roughly proportional to freeze-out dS  /dy. Therefore, ( dS  /dy)/N p 2/3 should be proportional to initial entropy density, a QGP signal. Initial Entropy Density: ~(dS  /dy)/Overlap Area Data indicates that the initial entropy density does grow with centrality, but not very rapidly. Solid envelope = Systematic errors in N p Our QGP “smoking gun” seems to be inhaling the smoke! STAR Preliminary

24. STAR October 16, 2003 John G. Cramer24 Conclusions 1.The source-averaged pion phase space density  f  is very high, in the low momentum region roughly 2  that observed at the CERN SPS for Pb+Pb at  S nn =17 GeV. 2.The pion entropy per particle S  /N  is very low, implying a significant pion chemical potential (   ~44 MeV) at freeze out. 3.The total pion entropy at midrapidity dS  /dy grows linearly with initial participant number N p, with a slope of ~6.6 entropy units per participant. (Why?? Is Nature telling us something?) 4.For central collisions at midrapidity, the entropy content of all pions is ~5  greater than that of all nucleons+antinucleons. 5.The initial entropy density increases with centrality, but forms a convex curve that shows no indication of the dramatic increase in entropy density expected with the onset of a quark- gluon plasma.

25. STAR October 16, 2003 John G. Cramer25 The End