1. Steffen A. BassDynamics of Hadronization #1 Steffen A. Bass Duke University & RIKEN-BNL Research Center The baryon puzzle at RHIC Recombination + Fragmentation Model Results: single particle observables, correlations & entropy Dynamics of Hadronization: Interplay of Fragmentation and Recombination  R.J. Fries, C. Nonaka, B. Mueller & S.A. Bass, PRL 90 202303 (2003)  R.J. Fries, C. Nonaka, B. Mueller & S.A. Bass, PRC 68 044902 (2003)  C. Nonaka, B. Mueller, M. Asakawa, S.A. Bass & R.J. Fries, PRC 69 031902 (2004)  R.J. Fries, S.A. Bass & B. Mueller, PRL in print (nucl-th/0407102)  C. Nonaka, B. Mueller, S.A. Bass & M. Asakawa, nucl-th/0501028

2. Steffen A. BassDynamics of Hadronization #2 Standard Model of Hadronization initial state pre-equilibrium QGP and hydrodynamic expansion hadronization hadronic phase and freeze-out low p t :  hadron yields & ratios fit with a SM:  spectra via Hydro: high p t :  pQCD is applicable, hadronization via fragmentation  a fast parton fragments via a color string: a  h+X  hadron spectrum is given by: ?

3. Steffen A. BassDynamics of Hadronization #3 The baryon puzzle @ RHIC where does the large proton over pion ratio at high p t come from? why do protons not exhibit the same suppression as pions?  fragmentation yields N p /N π <<1  fragmentation starts with a single fast parton: energy loss affects pions and protons in the same way! ratio of KKP fragmentation functions for p and π from u quarks

4. Steffen A. BassDynamics of Hadronization #4 Species dependent saturation of elliptic flow hyperon v 2 saturates later and higher than kaon v 2. same effect observed for protons and pions. at low p T the phenomenology seems better described in m T – m 0 than p T, indicating hydro scaling, yet scaling breaks down for high p T what drives the different p T scales for K S and Λ v 2 ?  novel mechanism of baryon formation?

5. Steffen A. BassDynamics of Hadronization #5 Recombination+Fragmentation Model basic assumptions: at low p t, the quarks and antiquark spectrum is thermal and they recombine into hadrons locally “at an instant”:  features of the parton spectrum are shifted to higher p t in the hadron spectrum at high p t, the parton spectrum is given by a pQCD power law, partons suffer jet energy loss and hadrons are formed via fragmentation of quarks and gluons

6. Steffen A. BassDynamics of Hadronization #6 Recombination: Pro’s & Con’s Pro’s: for exponential parton spectrum, recombination is more effective than fragmentation baryons are shifted to higher p t than mesons, for same quark distribution  understand behavior of protons! Con’s: simple recombination may violate entropy conservation gluons at hadronization need to be converted recombining partons: p 1 +p 2 =p h fragmenting parton: p h = z p, z<1

7. Steffen A. BassDynamics of Hadronization #7 Recombination: new life for an old idea High Energy Physics Phenomenology: K.P. Das & R.C. Hwa, Phys. Lett. B68, 459 (1977) Quark-Antiquark Recombination in the Fragmentation Region  description of leading particle effect T. Ochiai, Prog. Theo. Phys. 75, 1184 (1986) E. Braaten, Y. Jia & T. Mehen, Phys. Rev. Lett. 89, 122002 (2002) R. Rapp & E.V. Shuryak, Phys. Rev. D67, 074036 (2003) Heavy-Ion Phenomenology: T. S. Biro, P. Levai & J. Zimanyi, Phys. Lett. B347, 6 (1995) ALCOR: a dynamical model for hadronization  yields and ratios via counting of constituent quarks R.C. Hwa & C.B. Yang, Phys. Rev. C66, 025205 (2002) R. Fries, B. Mueller, C. Nonaka & S.A. Bass, Phys. Rev. Lett. 90 V. Greco, C.M. Ko and P. Levai, Phys. Rev. Lett. 90 Anisotropic flow: S. Voloshin, QM2002, Nucl. Phys. A715, 379 (2003) Z.W. Lin & C.M. Ko, Phys. Rev. Lett 89, 202302 (2002) D. Molnar & S. Voloshin, Phys. Rev. Lett 91, 092301 (2003)

8. Steffen A. BassDynamics of Hadronization #8 Recombination: nonrelativistic formalism use thermal quark spectrum given by: w(p) = exp(-p/T) for a Gaussian meson wave function with momentum width Λ M, the meson spectrum is obtained as: similarly for baryons:

9. Steffen A. BassDynamics of Hadronization #9 Recombination: relativistic formalism choose a hypersurface Σ for hadronization use local light cone coordinates (hadron defining the + axis) w a (r,p): single particle Wigner function for quarks at hadronization Ф M & Ф B : light-cone wave-functions for the meson & baryon respectively x, x’ & (1-x): momentum fractions carried by the quarks integrating out transverse degrees of freedom yields:

10. Steffen A. BassDynamics of Hadronization #10 Recombination vs. Fragmentation Fragmentation… … but it wins out at large p T, when the spectrum is a power law ~ (p T ) -b : … never competes with recombination for a thermal (exponential) spectrum:

11. Steffen A. BassDynamics of Hadronization #11 Hadron Spectra

12. Steffen A. BassDynamics of Hadronization #12 Hadron Ratios vs. p t

13. Steffen A. BassDynamics of Hadronization #13 Flavor Dependence of high-p t Suppression R+F model describes different R AA behavior of protons and pions in the fragmentation region all hadron flavors exhibit jet-quenching

14. Steffen A. BassDynamics of Hadronization #14 anisotropic or “elliptic” flow is sensitive to initial geometry Elliptic Flow more flow in collision plane than perpendicular to it less absorption in collision plane than perpendicular to it low p t domain:high p t domain: total elliptic flow is the sum of both contributions: r(p t ): relative weight of the fragmentation contribution in spectra

15. Steffen A. BassDynamics of Hadronization #15 Parton Number Scaling of v 2  smoking gun for recombination  measurement of partonic v 2 ! in leading order of v 2, recombination predicts:

16. Steffen A. BassDynamics of Hadronization #16 Resonance v 2 : scaling violations  QGP resonances: hadronizing QGP, no rescattering  HG resonances: hadronic phase, h-h rescattering Key: v 2 is additive for composite particles quarks n=2 scaling n=4 scaling Total is determined by experiments and related to width of particles and cross section in the hadronic medium.

17. Steffen A. BassDynamics of Hadronization #17 Two-Particle Correlations: a Challenge? PHENIX & STAR measure associated yields in p T windows of a few GeV/c. trigger hadron A, associated hadron B: associated yield as a function of relative azimuthal angle  clear jet-like structure observed at intermediate p T  very similar to p+p; jet fragmentation? analyze as function of integrated yield:  simple recombination of uncorrelated thermal quarks cannot reproduce two particle correlations

18. Steffen A. BassDynamics of Hadronization #18 Recombination: Inclusion of Correlations Recombination approach allows for two particle correlations, provided they are contained in the parton source distributions Three distinct types are conceivable: F-F, SH-F and SS-SS Ansatz for SS-SS: for two mesons, use product of correlated parton distributions:  Which results in a correlated two hadron yield:

19. Steffen A. BassDynamics of Hadronization #19 Correlations: Proof of Principle Meson-triggerBaryon-trigger  strong correlations from fragmentation, but suppressed by soft triggers  combination of hard fragmentation and soft recombination correlations with a fixed correlation volume is compatible with data

20. Steffen A. BassDynamics of Hadronization #20 Recombination: Entropy Puzzle Does recombination violate the 2nd law of thermodynamics ? –particle number decreases drastically in hadronization via reco…  restrict reco approach to intermediate momenta, ignore bulk…  decay of hadronic resonances as possible solution (Greco et al.)  need estimate of entropy at hadronization quark recombination hadronization

21. Steffen A. BassDynamics of Hadronization #21 1) resonance gas model: –massless particles: –massive particles: Entropy in the Hadronic Phase  entropy content is larger than often assumed! : resonances : ground state MeV : bosons : fermions 4.72 5.15 2) final state entropy from data: Pal & Pratt, PLB578,310 (2004) STAR: PRC68, 044905 (2003) Hydro+UrQMD calculations indicate a 5% increase in multiplicity due to rescattering in the hadronic phase

22. Steffen A. BassDynamics of Hadronization #22 Entropy in the Deconfined Phase Lattice QCD [CP-PACS with N t =6 & N f =2] systematic uncertainties include: thermodynamic limit, continuum limit, unphysically large quark mass…  entropy content of the deconfined phase near T C is strongly reduced due to interactions!

23. Steffen A. BassDynamics of Hadronization #23 Entropy Puzzle resolved? No direct comparison of the entropy content of both phases: – Volume at hadronization ? – Number of quarks on the lattice ? S H is larger than often assumed. S Q near T C is strongly reduced due to interactions. Recombination may be compatible with the entropy constraint after all! ? Quark PhaseHadron Phase

24. Steffen A. BassDynamics of Hadronization #24 Summary & Outlook The Recombination + Fragmentation Model: provides a natural solution to the baryon puzzle at RHIC describes the intermediate and high p t range of  hadron ratios & spectra  jet-quenching phenomena  elliptic flow  leading / next-to-leading particle correlations issues to be addressed in the future: treatment of gluons & higher Fock states (work in progress) realistic space-time dynamics of parton source need improved data of identified hadrons at high p t

25. Steffen A. BassDynamics of Hadronization #25 The End

26. Steffen A. BassDynamics of Hadronization #26 transverse momentum p t time Connecting the dots … initial state pre-equilibrium QGP and hydrodynamic expansion hadronization hadronic phase and freeze-out shattered color-glas jet production hydrodynamic evolution jet quenching parton recombination fragmentation reco/SM? radial flow HBT ?!

27. Steffen A. BassDynamics of Hadronization #27 Parton Number Scaling of Elliptic Flow in the recombination regime, meson and baryon v 2 can be obtained from the parton v 2 in the following way:  neglecting quadratic and cubic terms, one finds a simple scaling law: