Looking Through the “Veil of Hadronization”: Pion Entropy & PSD at RHIC

1. Looking Through the“Veil of Hadronization”:Pion Entropy & PSD at RHIC John G. CramerDepartment of PhysicsUniversity of Washington, Seattle, WA, USA STAR Collaboration MeetingCalifornia Institute of Technology February 18, 2004

2. 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 Dp3Dx3 = h3. • The source-averaged phase space density is áf(p)ñ º ∫[f(p, x)]2 d3x / ∫f(p, x) d3x, i.e., the local phase space density averaged over the f-weighted source volume. Because of Liouville’s Theorem, for free-streaming particles áf(p)ñ is a conserved Lorentz scalar. Sinyukov has recently asserted that áf(p)ñ 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. John G. Cramer

3. Entropy: Calculation & Expectations • Entropy – The pion entropy per particle Sp/Np and the total pion entropy at midrapidity dSp/dy can be calculated from áf(p)ñ. 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 (2nd Law of Thermodynamics.) 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 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. hep-ph/0212302 nucl-th/0104023 Entropy penetrates the “veil of hadronization”. John G. Cramer

4. Pion Phase Space Density at Midrapidity The source-averaged phase space density áf(mT)ñ is the dimensionless number of pions per 6-dimensional phase space cell h3, as averaged over the source. At midrapidity áf(mT)ñ is given by the expression: Average phasespace density HBT “momentumvolume” Vp PionPurityCorrection Momentum Spectrum Jacobianto make ita Lorentzscalar John G. Cramer

5. RHIC Collisions as Functions of Centrality Frequency of Charged Particlesproduced in RHIC Au+Au Collisions At RHIC we can classifycollision events by impact parameter, based on charged particle production. of sTotal Participants Binary Collisions John G. Cramer

6. Corrected HBT Momentum VolumeVp /l½ 50-80% Centrality 130 GeV/nucleon 40-50% Peripheral 30-40% Fits assuming: Vpl-½=A0 mT3a (Sinyukov) 20-30% 10-20% 5-10% 0-5% Central STAR Preliminary mT - mp (GeV) John G. Cramer

7. Global Fit to Pion Momentum Spectrum 130 GeV/nucleon We make a global fit of the uncorrected pion spectrum vs. centrality by: • Assuming that the spectrumhas the form of an effective-TBose-Einstein distribution: d2N/mTdmTdy=A/[Exp(E/T) –1] and • Assuming that A and T have aquadratic dependence on thenumber of participants Np:A(p) = A0+A1Np+A2Np2T(p) = T0+T1Np+T2Np2 STAR Preliminary John G. Cramer

8. Interpolated Phase Space Densityáfñat S½ = 130 GeV HBT points with interpolated spectra Note failure of “universal” PSDbetween CERN and RHIC. } NA49 Central STAR Preliminary Peripheral John G. Cramer

9. Extrapolated Phase Space Densityáfñat S½ = 130 GeV Spectrum points with extrapolated HBT Vp/l1/2 Central Note that for centralities of 0-40% of sT, áfñ changes very little. áfñdrops only for the lowest 3 centralities. STAR Preliminary Peripheral John G. Cramer

10. Converting áfñ to Entropy per Particle (1) Starting from quantum statistical mechanics, we define: +0.2% An estimate of the average pion entropy per particle áS/Nñ can be obtainedfrom a 6-dimensional space-momentum integral over the local phase spacedensity f(x,p): O(f) O(f3) O(f4) +0.1% dS6(Series)/dS6 1.000 To perform the space integrals, we assume that f(x,p) = áf(p)ñg(x),where g(x) = Ö23 Exp[-x2/2Rx2-y2/2Ry2-z2/2Rz2], i.e., that the source hasa Gaussian shape based on HBT analysis of the system. Further, we make theSinyukov-inspired assumption that the three radii have a momentum dependenceproportional to mT-a. Then the space integrals can be performed analytically.This gives the numerator and denominator integrands of the above expressionfactors of RxRyRz = Reff3mT-3a.(For reference, a~½) -0.1% O(f2) -0.2% f John G. Cramer

11. Converting áfñ to Entropy per Particle (2) The entropy per particle áS/Nñ then reduces to a momentum integralof the form: (6-D) (3-D) (1-D) We obtain a from the momentum dependence of Vpl-1/2 and performthe momentum integrals numerically using momentum-dependent fits to áfñor fits to Vpl-1/2 and the spectra. John G. Cramer

12. Entropy per Pion fromVp /l½and Spectrum Fits Peripheral STAR Preliminary Line = Combined fits to spectrum and Vp/l1/2 Central John G. Cramer

13. Thermal Bose-Einstein Entropy per Particle The thermal estimate of the p entropy per particle can beobtained by integrating a Bose-Einstein distribution over3D momentum: mp/mp T/mp mp= 0 mp= mp Note that the thermal-model entropy per particle usually decreases with increasing temperature T and chemical potential mp. John G. Cramer

14. Entropy per Particle S/N with Thermal Estimates STAR Preliminary Peripheral Solid line and points show S/Nfrom spectrum and Vp/l1/2 fits. For T=120 MeV, S/N impliesa pion chemical potential ofmp=63 MeV. Dashed line indicates systematicerror in extracting Vp from HBT. Central John G. Cramer

15. Total Pion Entropy dSp/dy STAR Preliminary Dashed line indicates systematicerror in extracting Vp from HBT. P&P Why is dSp/dylinear with Np?? Dot-dash line indicates dS/dy fromBSBEx fits to interpolated <f>. P&P Entropy content ofnucleons + antinucleons John G. Cramer

16. Total Pion Entropy per Participant (dSp/dy)/Np Central Average Peripheral John G. Cramer

17. Conclusions • The source-averaged pion phase space density áfñ is very high, in the low momentum region roughly 2´ that observed at the CERN SPS for Pb+Pb at ÖSnn=17 GeV. • The pion entropy per particle Sp/Np is very low, implying a significant pion chemical potential (mp~63 MeV) at freeze out. • For central collisions at midrapidity, the entropy content of all pions is ~5´ greater than that of all nucleons+antinucleons. • The total pion entropy at midrapidity dSp/dy grows linearly with initial participant number Np. (Why?? Is Nature telling us something?) • The pion entropy per participant (dSp/dy)/ Np , 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. • Our next priority is to obtain similar estimates of (dSp/dy)/ Npfor the d+Au and p+p systems at RHIC. John G. Cramer

18. The End John G. Cramer