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Pion Entropy and Phase Space Density at RHIC

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.

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Pion Entropy and Phase Space Density at RHIC

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  1. Pion Entropy andPhase Space Density at RHIC John G. CramerDepartment of PhysicsUniversity of Washington, Seattle, WA, USA Second Warsaw Meeting on Particle Correlations and Resonancesin Heavy Ion CollisionsWarsaw University of Technology October 16, 2003

  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 thef-weighted source volume. Because of Liouville’s Theorem, for free-streaming particles áf(p)ñ 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. 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. 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 Can Entropy provide the QGP “Smoking Gun”?? 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. 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 (Snn½ = 130 GeV) into 7 centrality bins for |y| < 0.5, incorporating several improvements : • We use 6 KT bins (average pair momentum) rather than 3 pT bins (individual pion momentum) for pair correlations (better large-Q statistics). • 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. • We do event mixing only for events in ±300 of the same reaction plane. • We combined p+p+ and p-p- correlations (improved statistics). • 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 reducingl, 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 <f> for the more peripheral centralities. This bug had no effect on the 0-5% centrality. John G. Cramer

  6. 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

  7. Corrected HBT Momentum Volume Vp /l½ 50-80% Centrality 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

  8. Global Fit to Pion Momentum Spectrum 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

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

  10. Extrapolated Pion Phase Space Density áfñat 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

  11. Converting Phase Space Density 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

  12. Converting Phase Space Density 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

  13. Blue-Shifted Bose-Einstein Functions To integrate over the phase space density, we need a function of pT 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: <f>Static = {Exp[(mTotal - mp)/T0] - 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 transverseflow rapidities hL and hT, the average values <hL> and <hT> to obtain: We assume mp=<hL>=0 and consider three models for <hT>: BSBE1: <hT> = a(i.e., constant average flow, independent of pT) BSBE2: <hT> = a (pT/mT) = a bT(i.e., proportional to pair velocity) BSBE3: <hT> = a1bT +a3bT3+a5bT5+a7bT7(minimize D(S/N)/D flow) John G. Cramer

  14. Fits to Interpolated Pion Phase Space Density HBT points with interpolated spectra Fitted with BSBE2 function Central STAR Preliminary Warning: PSD in the region measured contributes only about 60% to the average entropy per particle. Peripheral John G. Cramer

  15. Fits to Extrapolated Pion Phase Space Density Solid = Combined Vp/l1/2 and Spectrum fits Dashed = Fitted with BSBE2 function Central STAR Preliminary Spectrum points with extrapolated HBT Vp/l1/2 Each successive centrality reduced by 3/2 Peripheral John G. Cramer

  16. Large-mT behavior of three BSBE Models Solid = BSBE2: hT = a bT Dotted = BSBE3: 7th order odd polynomial in bT Dashed = BSBE1: hT = Constant Each successive centrality reduced by 3/2 John G. Cramer

  17. Large mT behavior using Radius & Spectrum Fits Solid = fits to spectrum and Vp/l1/2 Dashed = BSBE2 fits to extrapolated data Each successive centrality reduced by 3/2 John G. Cramer

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

  19. Entropy per Pion from BSBE Fits Peripheral STAR Preliminary Black = Combined fits to spectrum and Vp/l1/2 Red = BSBE1: Const Green = BSBE2:~ bT Blue = BSBE3: Odd 7th order Polynomial in bT Central John G. Cramer

  20. 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

  21. 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=110 MeV, S/N impliesa pion chemical potential ofmp=44.4 MeV. Dashed line indicates systematicerror in extracting Vp from HBT. Central Dot-dash line shows S/N from BDBE2 fits to áfñ John G. Cramer

  22. Total Pion Entropy dSp/dy STAR Preliminary Dashed line indicates systematicerror in extracting Vp from HBT. P&P Why is dSp/dylinear with Np?? Solid line is a linear fit through (0,0)with slope = 6.58 entropy unitsper participant Dot-dash line indicates dS/dy fromBSBEx fits to interpolated <f>. P&P Entropy content ofnucleons + antinucleons John G. Cramer

  23. Initial Entropy Density: ~(dSp/dy)/Overlap Area Initial collision overlap area is roughlyproportional to Np2/3 Initial collision entropy is roughlyproportional to freeze-out dSp/dy. Therefore, (dSp/dy)/Np2/3should be proportionalto initial entropydensity, a QGPsignal. Solid envelope =Systematic errors in Np STAR Preliminary Data indicates that the initialentropy density does grow withcentrality, but not very rapidly. Our QGP “smoking gun” seems to beinhaling the smoke! John G. Cramer

  24. 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~44 MeV) at freeze out. • The total pion entropy at midrapidity dSp/dy grows linearly with initial participant number Np, with a slope of ~6.6 entropy units per participant. (Why?? Is Nature telling us something?) • For central collisions at midrapidity, the entropy content of all pions is ~5´ greater than that of all nucleons+antinucleons. • 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. John G. Cramer

  25. The End John G. Cramer

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