1. Solving the RHIC HBT Puzzle John G. Cramer and Gerald A. Miller University of Washington Seattle, Washington John G. Cramer and Gerald A. Miller University of Washington Seattle, Washington Femtoscopy @ RHIC/AGS Users Meeting Brookhaven National Laboratory June 21, 2005

2. RHIC/AGS Users Meeting2 Outline of Talk Part 1 – Introduction (Cramer) RHIC Physics and the HBT Puzzle Overview of our model Part 2 – Theory and Formalism (Miller) Distorted Waves and the Emission Function The Optical Potential and Chiral Symmetry Opacity and Refraction Part 3 – Implementation and Fits to Data (Cramer) HBT Radius Fits Spectrum Fits Ramsauer Resonances Summary Outlook

3. Part 1 Introduction John G. Cramer Introduction John G. Cramer

4. June 21, 2005RHIC/AGS Users Meeting4 The RHIC HBT Puzzle The data from the first four years of RHIC operation paint a confusing picture. Some evidence supports the presence of a QGP in the early stages of Au+Au collisions: There is evidence that relativistic hydrodynamics works very well in describing the low and medium energy dynamics of the collision, suggesting a fluid-like medium. There is evidence from elliptic flow data of very high initial pressure and collective behavior.

5. June 21, 2005RHIC/AGS Users Meeting5 The Featureless HBT Landscape The source radii, as inferred from HBT interferometry, are very similar over almost two orders of magnitude in collision energy. The ratio of R o /R s is near 1 at all energies, which naively implies a “hard” equation of state and explosive emission behavior. AGS CERN RHIC  

6. June 21, 2005RHIC/AGS Users Meeting6 Kolb and U. Heinz (2002) Hydrodynamics and Elliptic Flow Elliptic flow (V 2 ) provides evidence of very large initial pressure. Relativistic hydrodynamics works very well in explaining this and the particle spectra (Examples:   and p)

7. June 21, 2005RHIC/AGS Users Meeting7 The RHIC HBT Puzzle The data from the first four years of RHIC operation paint a confusing picture. Some evidence supports the presence of a QGP in the early stages of Au+Au collisions: There is evidence that relativistic hydrodynamics works very well in describing the low and medium energy dynamics of the collision, suggesting a fluid-like medium. There is evidence from elliptic flow data of very high initial pressure and collective behavior. There is evidence of strong suppression of the most energetic pions, those that should be produced in the early stages of the collision. There is evidence of strong suppression of back-to-back jets.

8. June 21, 2005RHIC/AGS Users Meeting8 PRL91 072305 (2003) d+Au PRL91 072304 (2003) Suppression of High p T Particles High momentum particles from the initial stages of the collision are suppressed, both in particle spectra and in back-to-back jets.

9. June 21, 2005RHIC/AGS Users Meeting9 The RHIC HBT Puzzle The data from the first four years of RHIC operation paint a confusing picture. Some evidence supports the presence of a QGP in the early stages of Au+Au collisions: There is evidence that relativistic hydrodynamics works very well in describing the low and medium energy dynamics of the collision, suggesting a fluid-like medium. There is evidence from elliptic flow data of very high initial pressure and collective behavior. There is evidence of strong suppression of the most energetic pions, those that should be produced in the early stages of the collision. There is evidence of strong suppression of back-to-back jets. BUT … a QGP-driven Au+Au system should expand to a fairly large size and should show a fairly long duration of pion emission. However, inteferometry says otherwise:  HBT interferometry analysis indicates that the Au+Au collisions at RHIC seem to be about the same size as collisions at much lower energies at the SPS and AGS.  HBT interferometry analysis indicates that the emission of pions is of very short duration, less than 1 fm/c, so short that a duration can’t be extracted from data. This explosive behavior would imply a very “hard” equation of state (EOS) for the system, while the QGP EOS is “soft” because of the many degrees of freedom.

10. June 21, 2005RHIC/AGS Users Meeting10 The RHIC HBT Puzzle The data from the first four years of RHIC operation paint a confusing picture. Some evidence supports the presence of a QGP in the early stages of Au+Au collisions: There is evidence that relativistic hydrodynamics works very well in describing the low and medium energy dynamics of the collision, suggesting a fluid-like medium. There is evidence from elliptic flow data of very high initial pressure and collective behavior. There is evidence of strong suppression of the most energetic pions, those that should be produced in the early stages of the collision. There is evidence of strong suppression of back-to-back jets. BUT … a QGP-driven Au+Au system should expand to a fairly large size and should show a fairly long duration of pion emission. However, inteferometry says otherwise:  HBT interferometry analysis indicates that the Au+Au collisions at RHIC seem to be about the same size as collisions at much lower energies at the SPS and AGS.  HBT interferometry analysis indicates that the emission of pions is of very short duration, less than 1 fm/c, so short that a duration can’t be extracted from data. This explosive behavior would imply a very “hard” equation of state (EOS) for the system, while the QGP EOS is “soft” because of the many degrees of freedom. That is the RHIC HBT Puzzle. Instead of bringing the nuclear liquid to a gentle boil and observing the steam of a QGP, the whole boiler seems to be exploding in our face!

11. June 21, 2005RHIC/AGS Users Meeting11 Overview of Our 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 due to chiral-symmetry breaking as they pass from the hot, dense collision medium [m(  )  0]) to the outside vacuum [m(  )  140 MeV]. This is accomplished by solving the Klein-Gordon equation with an optical potential, the real part of which is a deep, attractive “mass-type” potential. We use relativistic quantum mechanics in a partial wave expansion to treat the behavior of the pions used in the HBT analysis. We note that most RHIC theories have been semi-classical, even though HBT analysis uses pions in the momentum region (p  < 600 MeV/c) where quantum wave-mechanical effects should be important. The model calculates only the spectrum of particles participating in the HBT correlation (not the spectrum from long-lived “halo” resonances).

12. June 21, 2005RHIC/AGS Users Meeting12 q out q side q long R side R long R out p1p1 p2p2 p2p2 + p2p2 p1p1 q Quantum mechanical interference - space time separation of source. q = p 1  p 2 K = ½ (p 1  p 2 ) C(q,K)  1   p 1,p 2      p 1  p 2  1   1  q 2 L R 2 L  q 2 S R 2 S  q 2 O R 2 O  …  HBT 2-Particle Interferometry Hydrodynamics predicts big R O /R S, Data says R O /R S about 1 HBT puzzle

13. June 21, 2005RHIC/AGS Users Meeting13 About Chiral Symmetry Question 1: The up and down “current” quarks have masses of 5 to 10 MeV. The   (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   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. medium vacuum

14. Part 2 Formalism and Theory Gerald A. Miller Formalism and Theory Gerald A. Miller

15. June 21, 2005RHIC/AGS Users Meeting15 Formalism Wigner distribution of  source current density matrix S 0 (x,K) Pions interact with dense medium  is distorted (not plane) wave Gyulassy et al ‘79 chaotic sources

16. June 21, 2005RHIC/AGS Users Meeting16 Source Properties (“hydrodynamics inspired” source function of Heinz & collaborators) (Bose-Einstein thermal function) (medium density)

17. June 21, 2005RHIC/AGS Users Meeting17 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

18. June 21, 2005RHIC/AGS Users Meeting18 The Meaning of U Im (U) : Opacity, Re (U) :Refraction pions lose energy and flux Re(U) must exist: very strong attraction chiral phase transition Im[U 0 ]=-p   0,  1 mb,    = 1fm -3, Im[U 0 ] = .15 fm -2,  = 7 fm

19. June 21, 2005RHIC/AGS Users Meeting19 Son & Stephanov 2002 v 2, v 2 m 2   approach  near T = T c Both terms of U are negative (attractive) U(b)=-(w 0 +w 2 p 2 )  (b), w 0 =real, w 2 =complex

20. June 21, 2005RHIC/AGS Users Meeting20 Compute Correlation Function Correlation function is not Gaussian; we evaluate it near the q of experiment. The R 2 values are not the moments of the emission function S.

21. June 21, 2005RHIC/AGS Users Meeting21 Semi-Classical Eikonal Opacity b l R Heiselberg and Vischer X +

22. June 21, 2005RHIC/AGS Users Meeting22 Influence of the Real Potential in the Eikonal Approximation Therefore the real part of U, no matter how large, has no influence here.

23. June 21, 2005RHIC/AGS Users Meeting23 Source De-magnification by the Real Potential Well 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. n=1.00 n=1.33 A Fly in a Bubble Rays bend closer to radii V csr = (120 MeV) 2 Velocity in well Velocity in vacuum

24. June 21, 2005RHIC/AGS Users Meeting24 |  ( , b)|   b ) at K T = 1.000 fm -1 = 197 MeV/c Wave Function of Full Calculation Imaginary Only Eikonal Observer

25. June 21, 2005RHIC/AGS Users Meeting25 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.) Sone time-dependent effects can be manifested in the energy-dependence of the optical potential. (Time and energy are conjugate quantum variables.) The optical potential also includes the effects of resonances, including the heavy ones. Therefore, our present treatment implicitly includes those resonances produced inside the medium. However, a more detailed coupled-channels calculation could be done, in which selected resonances were treated as explicit channels. Describing the present STAR data apparently does not require such an elaboration.

26. Part 3 Implementation John G. Cramer Implementation John G. Cramer

27. June 21, 2005RHIC/AGS Users Meeting27 Fitting 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. The model uses 8 pion source parameters and 3 optical potential parameters, for a total of 11 parameters in the model. We have fitted STAR data at  s NN =200 GeV, simultaneously fitting R o, R s, R l, and dN p /dy (fitting both magnitude and shape) at 8 momentum values (i.e., 32 data points), using a Levenberg- Marquardt fitting algorithm. In the resulting fit, the  2 per data point is ~2.2 and the  2 per degree of freedom is ~3.3. We remove long-lived “halo” resonance contributions to the spectrum (which are not included in the model) by multiplying the uncorrected spectrum by ½ (the HBT parameter) before fitting.

28. June 21, 2005RHIC/AGS Users Meeting28 Fits to 200 GeV Pion HBT Radii U=0 Re[U]=0 No flow Boltzmann Full Calculation Non-solid curves show the effects of turning off various parts of the calculation.

29. June 21, 2005RHIC/AGS Users Meeting29 Fit to 200 GeV Pion Spectrum U=0 Re[U]=0 No flow Boltzmann Full Calculation Raw Fit Non-solid curves show the effects of turning off various parts of the calculation

30. June 21, 2005RHIC/AGS Users Meeting30 Meaning of the Parameters Temperature: 222 MeV  Chiral PT predicted at ~ 170 MeV Transverse flow rapidity: 1.6  v max = 0.93 c, v av = 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 p 2  deep well  strong attraction. Im(U): 0.128 p 2  mfp  8 fm @ K T =1 fm -1  strong absorption  high density Pion chemical potential: m  =124 MeV, slightly less than mass(  ) We have evidence for a CHIRAL PHASE TRANSITION!

31. June 21, 2005RHIC/AGS Users Meeting31 Low p T Ramsauer Resonances R O (fm) Pion Spectrum K T (MeV/c) R S (fm) Phobos (corrected) Raw Fit U=0 Re[U]=0 Boltzmann No flow Full Calculation |  (q, b)| 2  (b) at K T = 49.3 MeV/c

32. June 21, 2005RHIC/AGS Users Meeting32 Potential-Off Radius Fits No Chemical or Optical Pot. No Optical No Real STAR Blast Wave Full Calculation Non-solid curves show the effects of refitting. Out Side LongR O /R S Ratio

33. June 21, 2005RHIC/AGS Users Meeting33 Potential-Off Spectrum Fits No Chemical or Optical Pot. No Real No Optical STAR Blast Wave Full Calculation Raw Fit Non-solid curves show the effects of potential- off refits.

34. June 21, 2005RHIC/AGS Users Meeting34 Summary Quantum mechanics has solved the technical problems of applying opacity to HBT. We obtain excellent fits to STAR  s NN =200 GeV data, simultaneously fitting three HBT radii and the p T 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 for 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.

35. June 21, 2005RHIC/AGS Users Meeting35 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 (K T ~150 MeV/c). We are presently attempting to “track” the CPT phenomenon to lower collision energies, where the deep real potential should not be present. A detailed paper for Phys. Rev. C describing this distorted wave emission function theory and its implementation will be placed on the ArXiv preprint server soon.

36. The End A paper describing this work has been published in Phys. Rev. Lett. 94, 102302 (2005), and is on the ArXiv preprint server as nucl-th/0411031