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The RHIC HBT Puzzle, Chiral Symmetry Restoration, and Pion Opacity John G. Cramer (with Gerald A. Miller) University of Washington Seattle, Washington, USA Quark Matter 2005 Budapest August 5, 2005 The Featureless HBT Landscape The source radii, as inferred from HBT interferometry, are very similar over almost two orders of magnitude in collision energy. AGS CERN RHIC The ratio of Ro/Rs is near 1 at all energies, which naively implies a “hard” equation of state and explosive emission behavior. August 5, 2005 Quark Matter 2005, Budapest 2 HBT Momentum Geometry Relative momentum between pions is a vector q p1 p 2 can extract 3D shape information Rlong – along beam direction Rout – along “line of sight”, includes time/energy information. Rside – “line of sight”, no time/energy information. K 12 p1 p2 p1 q Rside Rout p2 August 5, 2005 Quark Matter 2005, Budapest 3 Overview of the DWEF Model The medium is dense and strongly interacting, so the pions must “fight” their way out to the vacuum. This modifies their wave functions, producing the distorted waves used in the model. We explicitly treat the absorption of pions by inelastic processes (e.g., quark exchange and rearrangement) as they pass through the medium, as implemented with the imaginary part of an optical potential. We explicitly treat the mass-change of pions (e.g., due to chiralsymmetry breaking) as they pass from the hot, dense collision medium [m(p)0]) to the outside vacuum [m(p)140 MeV]. This is accomplished by solving the Klein-Gordon equation with an optical potential, the real part of which is a deep, attractive, momentumdependent “mass-type” potential. We use relativistic quantum mechanics in a cylindrical geometry partial wave expansion to treat the behavior of pions producing Bose-Einstein correlations. We note that most RHIC theories are semi-classical, even though most HBT analyses use pions in the momentum region (pp < 600 MeV/c) where quantum wave-mechanical effects should be important. The model calculates only the spectrum of pions participating in the BE correlation (not those contributions to the spectrum from long-lived “halo” resonances, etc.). August 5, 2005 Quark Matter 2005, Budapest 4 About Chiral Symmetry Question 1: The up and down “current” quarks have masses of 5 to 10 MeV. The p (a down + anti-up combination) has a mass of ~140 MeV. Where does the observed mass come from? Answer 1: The quarks are more massive in vacuum due to “dressing”. Also the pair is tightly bound by the color force into a particle so small that quantum-uncertainty zitterbewegung gives both quarks large average momenta. Part of the p mass comes from the kinetic energy of the constituent quarks . Question 2: What happens when a pion is placed in a hot, dense medium? Answer 2: Two things happen: 1. The binding is reduced and the pion system expands because of external color forces, reducing the zitterbewegung and the pion mass. 2. The quarks that were “dressed” in vacuum become “undressed” in medium, causing up, down, and strange quarks to become more similar and closer to massless particles, an effect called “chiral symmetry restoration”. In many theoretical scenarios, chiral symmetry restoration and the quark-gluon plasma phase go together. Question 3: How can a pion regain its mass when it goes from medium to vacuum? Answer 3: It must do work against an average attractive force, losing kinetic energy while gaining mass. In effect, it must climb out of a potential well ~140 MeV deep. August 5, 2005 Quark Matter 2005, Budapest vacuum medium 5 Time-Independence, Resonances, and Freeze-Out We note that our use of a time-independent optical potential does not invoke the mean field approximation and is formally correct according to quantum scattering theory. (The semi-classical mind-set can be misleading.) While the optical potential is not time-dependent, some timedependent effects can be manifested in the energy-dependence of the potential . (Time and energy are conjugate quantum variables.) An optical potential can implicitly include the effects of resonances, including heavy ones. Therefore, our present treatment implicitly includes resonances produced within the hot, dense medium. We note that more detailed quantum coupled-channels calculations could be done, in which selected resonances were treated as explicit channels coupled through interactions. Describing the present STAR data apparently does not require this kind of elaboration. August 5, 2005 Quark Matter 2005, Budapest 6 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 8 pion source parameters and 3 optical potential parameters, for a total of 11 parameters in the model. The correlation function C (not the 2nd moment of C) is calculated. We have fitted STAR data at sNN=200 GeV, simultaneously fitting Ro, Rs, Rl, and dNp/dy (fitting both magnitude and shape) at 8 momentum values (i.e., 32 data points), using a LevenbergMarquardt fitting algorithm. In the resulting fit, the c2 per data point is ~2.2 and the c2 per degree of freedom is ~3.3. Only statistical (not systematic) errors are used in calculating c2. We remove long-lived “halo” resonance contributions to the spectrum (which are not included in the model) by multiplying the uncorrected spectrum by l½ (the HBT parameter) before fitting, then “un-correcting” the predicted spectrum with l½. August 5, 2005 Quark Matter 2005, Budapest 7 DWEF Fits to STAR 200 GeV Pion HBT Radii Full Calculation U=0 Boltzmann Re[U]=0 Non-solid curves show the effects of turning off various parts of the calculation. No flow August 5, 2005 Quark Matter 2005, Budapest 8 DWEF Fit to STAR 200 GeV Pion Spectrum Raw Fit Non-solid curves show the effects of turning off various parts of the calculation Full Calculation U=0 Boltzmann Re[U]=0 No flow August 5, 2005 Quark Matter 2005, Budapest 9 Meaning of the Parameters Temperature: 222 MeV; Chiral PT predicted at ~ 193 MeV Transverse flow rapidity: 1.6 vmax= 0.93 c, vav= 0.66 c Mean expansion time: 8.1 fm/c system expansion at ~ 0.5 c Pion emission between 5.5 fm/c and 10.8 fm/c soft EOS . WS radius: 12.0 fm = R(Au) + 4.6 fm > R @ SPS WS diffuseness: 0.72 fm (similar to Low Energy NP experience) Re(U): 0.113 + 0.725 p2 deep well strong attraction. Im(U): 0.128 p2 lmfp 8 fm @ KT=1 fm-1 strong absorption high density Pion chemical potential: mp=124 MeV, slightly less than mass(p) We have evidence suggesting a CHIRAL PHASE TRANSITION! August 5, 2005 Quark Matter 2005, Budapest 10 Low pT Ramsauer Resonances 14 16 12 14 RS 12 (fm) 10 RO 8 (fm) No flow 10 6 Boltzmann 8 4 2 6 KT (MeV/c) 10 20 30 40 50 60 70 KT (MeV/c) 10 20 30 40 50 Re[U]=0 70 Pion Spectrum Full Calculation |y(q, b)|2 r(b) at KT = 49.3 MeV/c 60 U=0 1.5 1 0.5 1 0 1 0.5 Raw Fit 0.5 0 For fit, would need l=0.411 KT (MeV/c) 0 -0.5 -0.5 -1 August 5, 2005 Phobos 0-6% (preliminary) -1 Quark Matter 2005, Budapest 11 Potential-Off Radius Fits Full Calculation Out Side No Real Non-solid curves show the effects of refitting. STAR Blast Wave RO/RS Ratio Long No Optical No Chemical or Optical Pot. August 5, 2005 Quark Matter 2005, Budapest 12 Potential-Off Spectrum Fits Model No Optical No Real Chi^2 Full Calculation Chi^2/#data Chi^2/#dof 69.19 2.16 3.29 905.09 28.28 43.10 No Optical Potential 1003.44 31.36 47.78 No Opt/ Chem Potential 1416.66 44.27 67.46 No Real Potential Raw Fit Non-solid curves show the effects of potentialoff refits. Full Calculation STAR Blast Wave No Chemical or Optical Pot. August 5, 2005 Quark Matter 2005, Budapest 13 200 GeV Cu+Cu Predictions 5.5 Scale RWS, t by Relect 5 5 4.5 4 Scale RWS, t by Scale RWS, AWS, t, Dt by Relect A1/3 RS fm RO fm 5.5 3.5 3 4.5 4 3.5 3 Scale RWS, AWS, t, Dt by A1/3 100 200 300 400 500 600 KT MeV c 7 1.3 6 1.2 RO RS RL fm 100 200 300 400 500 600 KT MeV c 5 1.1 4 1 3 0.9 100 200 300 400 500 600 KT MeV c August 5, 2005 Quark Matter 2005, Budapest STAR (preliminary) 100 200 300 400 500 600 KT MeV c 14 Summary Quantum mechanics has solved the technical problems of applying opacity to HBT. We obtain excellent DWEF fits to STAR sNN=200 GeV data, simultaneously fitting three HBT radii and the pT spectrum. The fit parameters are reasonable and indicate strong collective flow, significant opacity, and huge attraction. They describe pion emission in hot, highly dense matter with a soft pion equation of state. We have replaced the RHIC HBT Puzzle with evidence suggesting a chiral phase transition in RHIC collisions. We note that in most quark-matter scenarios, the QGP phase transition is accompanied by a chiral phase transition at about the same critical temperature. August 5, 2005 Quark Matter 2005, Budapest 15 Outlook We have a new tool for investigating the presence (or absence) of chiral phase transitions in heavy ion collisions. Its use requires both high quality pion spectra and high quality HBT analysis over a region that extends to fairly low momenta (KT~150 MeV/c). We are presently attempting to “track” the CPT phenomenon to lower collision energies, where the deep real potential should presumably go away. We plan to try to replace the empirical emission function with a relativistic hydrodynamic calculation of the multidimensional phase space density. (DWEF DWRHD) August 5, 2005 Quark Matter 2005, Budapest 16 The End A paper (with erratum) describing this work has been published in Phys. Rev. Lett. 94, 102302 (2005); See ArXiv: nucl-th/0411031; A longer paper has been submitted to PRC; See nucl-th/0507004 Backup Slides Formalism • Wigner distribution of p source current density matrix S0(x,K) • Pions interact with dense medium Gyulassy et al ‘79 is distorted (not plane) wave chaotic sources August 5, 2005 Quark Matter 2005, Budapest 19 Source Properties S0 ( x, k ) S0 (t , ) B (b, KT ) /(2p )3 2 2 (t t 0 ) cosh S 0 (t , ) exp 2 2 2 2 D 2p (Dt ) 2 Dt (“hydrodynamics inspired” source function of Heinz & collaborators) 1 B (b, KT ) M T r (b) K u p (medium density) exp 1 T 2 2 2 t t z (Bose-Einstein thermal function) tz 1 2 ln tz August 5, 2005 K particle momentum 4-vector u trasverse flow 4-vector Quark Matter 2005, Budapest 20 Wave Equation Solutions We assume an infinitely long Bjorken tube and azimuthal symmetry, so that the (incoming) waves factorize: 3D 2D(distorted)1D(plane) We solve the reduced Klein-Gordon wave equation: Partial wave expansion ! ordinary diff eq August 5, 2005 Quark Matter 2005, Budapest 21 The Meaning of U Im (U) : Opacity, Re (U) :Refraction pions lose energy and flux Im[U0]=-p r0, 1 mb, r0 = 1fm-3, Im[U0] = .15 fm-2, l = 7 fm Re(U) must exist: very strong attraction chiral phase transition August 5, 2005 Quark Matter 2005, Budapest 22 Son & Stephanov 2002 v2, v2 m2p Tapproach 0 near T = Tc Both terms of U are negative (attractive) U(b)=-(w0+w2p2)r(b), w0=real, w2=complex August 5, 2005 Quark Matter 2005, Budapest 23 Compute Correlation Function C ( K , q) 1 d d 4 4 2 x S ( x, K , q ) x S ( x, p1 ) d x S ( x, p2 ) 4 Correlation function is not Gaussian; we evaluate it near the q of experiment. The R2 values are not the moments of the emission function S. August 5, 2005 Quark Matter 2005, Budapest 24 Semi-Classical Eikonal Opacity b X l + R Heiselberg and Vischer August 5, 2005 Quark Matter 2005, Budapest 25 Influence of the Real Potential in the Eikonal Approximation Factors of cancel out in the product y (-) ( p1 , b)y *(-) ( p2 , b). Therefore the real part of U, no matter how large, has no influence here. August 5, 2005 Quark Matter 2005, Budapest 26 Source De-magnification by the Real Potential Well Velocity in well n=1.33 n=1.00 Velocity in vacuum Vcsr = (120 MeV)2 Because of the mass loss in the potential well, the pions move faster there (red) than in vacuum (blue). This de-magnifies the image of the source, so that it will appear to be smaller in HBT measurements. This effect is largest at low momentum. August 5, 2005 Quark Matter 2005, Budapest Rays bend closer to radii A Fly in a Bubble 27 |y(q, b)|2rb) at KT = 1.000 fm-1 = 197 MeV/c Observer 1 0.75 0.5 0.25 1 0 1 0.5 0.5 0 0 1 -0.5 0.5 1 -0.5 -1 Imaginary Only -1 0 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 Wave Function of Full Calculation Eikonal August 5, 2005 Quark Matter 2005, Budapest 28