1. STAR Looking Through the “Veil of Hadronization”: Pion Entropy & PSD at RHIC John G. Cramer Department of Physics University of Washington, Seattle, WA, USA STAR Collaboration Meeting California Institute of Technology February 18, 2004 STAR Collaboration Meeting California Institute of Technology February 18, 2004

2. STAR February 18, 2004 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. Sinyukov has recently asserted that  f(p)  is also approximately conserved from the initial collision to freeze out. 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 February 18, 2004 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. It never decreases (2 nd Law of Thermodynamics.) 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. Entropy penetrates the “veil of hadronization”.

4. STAR February 18, 2004 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 February 18, 2004 John G. Cramer5 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

6. STAR February 18, 2004 John G. Cramer6 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) 130 GeV/nucleon

7. STAR February 18, 2004 John G. Cramer7 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 130 GeV/nucleon

8. STAR February 18, 2004 John G. Cramer8 Interpolated 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

9. STAR February 18, 2004 John G. Cramer9 Extrapolated 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.

10. STAR February 18, 2004 John G. Cramer10 Converting  f  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% 

11. STAR February 18, 2004 John G. Cramer11 Converting  f  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)

12. STAR February 18, 2004 John G. Cramer12 Entropy per Pion from V p / ½ and Spectrum Fits Central Peripheral STAR Preliminary Line = Combined fits to spectrum and V p / 1/2

13. STAR February 18, 2004 John G. Cramer13 Thermal Bose-Einstein Entropy per Particle   = 0   = m  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  .

14. STAR February 18, 2004 John G. Cramer14 Entropy per Particle S/N with Thermal Estimates Central Peripheral STAR Preliminary Dashed line indicates systematic error in extracting V p from HBT. Solid line and points show S/N from spectrum and V p / 1/2 fits. For T=120 MeV, S/N implies a pion chemical potential of   =63 MeV.

15. STAR February 18, 2004 John G. Cramer15 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. Entropy content of nucleons + antinucleons P&P Why is dS  /dy linear with N p ??

16. STAR February 18, 2004 John G. Cramer16 Total Pion Entropy per Participant (dS  /dy)/N p Central Peripheral Average

17. STAR February 18, 2004 John G. Cramer17 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 (   ~63 MeV) at freeze out. 3.For central collisions at midrapidity, the entropy content of all pions is ~5  greater than that of all nucleons+antinucleons. 4.The total pion entropy at midrapidity dS  /dy grows linearly with initial participant number N p. (Why?? Is Nature telling us something?) 5.The pion entropy per participant (dS  /dy)/ N p, which should penetrate the “ veil of hadronization”, has a roughly constant value of 6.5 and shows no indication of the increase expected with the onset of a quark- gluon plasma. 6.Our next priority is to obtain similar estimates of (dS  /dy)/ N p for the d+Au and p+p systems at RHIC.

18. STAR February 18, 2004 John G. Cramer18 The End