1. Steffen A. BassHadronization @ RHIC #1 Steffen A. Bass Duke University & RIKEN-BNL Research Center Data: Protons at RHIC - the demise of pQCD? Recombination + Fragmentation Model Results and Predictions Hadronization @ RHIC: 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, nucl-th/0306027

2. Steffen A. BassHadronization @ RHIC #2 Some of the RHIC Puzzle(s): protons and the demise of pQCD? spezies-dependent saturation of v 2

3. Steffen A. BassHadronization @ RHIC #3 The proton 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. BassHadronization @ RHIC #4 Elliptic flow of K 0 and  hyperon v 2 saturates later and higher than kaon v 2. same effect observed for protons and pions. the phenomenology seems better described in m T – m 0 than p T ; why (kinetic energy)? what drives the different p T scales for K S and Λ v 2 ?  novel mechanism of baryon formation?

5. Steffen A. BassHadronization @ RHIC #5 A possible solution to the puzzle:  parton recombination

6. Steffen A. BassHadronization @ RHIC #6 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

7. Steffen A. BassHadronization @ RHIC #7 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: recombination violates 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

8. Steffen A. BassHadronization @ RHIC #8 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-ex/020014 Z.W. Lin & C.M. Ko, Phys. Rev. Lett 89, 202302 (2002) D. Molnar & S. Voloshin, nucl-th/0302014

9. Steffen A. BassHadronization @ RHIC #9 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:

10. Steffen A. BassHadronization @ RHIC #10 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:

11. Steffen A. BassHadronization @ RHIC #11 Recombination of thermal quarks results are insensitive to the model used for recombination (light- cone & Wigner functions vs. non-relativistic approx.) important features: p h = Σ p q d 3 N/dp 3 h  (w q ) n (with n=2,3) for a thermal distribution:  product of all Wigner functions only depends on hadron momentum!  Baryon/Meson ratio is independent of momentum, e.g. (C p, C π : degeneracy factors)

12. Steffen A. BassHadronization @ RHIC #12 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:

13. Steffen A. BassHadronization @ RHIC #13 p t range of parton recombination quark recombination (coalescence) may dominate for all p t < p 0. Combinatorical models (ALCOR, etc.) work well for total particle yields at SPS and RHIC low p t is not calculable, but calculation at moderate p t (few GeV) may be possible using hadron light-cone formalism transition from dense medium to dilute medium appears very rapid for fast partons (Δt=Δx/γ), validating sudden approximation focus on “high” p t evades problems of energy and entropy conservation in recombination: E = (p 2 +m 2 ) 1/2  p

14. Steffen A. BassHadronization @ RHIC #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. BassHadronization @ RHIC #15 Elliptic Flow: partons at low p t azimuthal anisotropy of parton spectra is determined by elliptic flow: with Blastwave parametrization for parton spectra: (Ф p : azimuthal angle in p-space) azimuthal anisotropy is parameterized in coordinate space and is damped as a function of p t :

16. Steffen A. BassHadronization @ RHIC #16 Elliptic Flow: partons at high p t azimuthal anisotropy is driven by parton energy/momentum loss Δp t R L r  L: average thickness of the medium α(p t ): damping function for low p t the unquenched parton p t distribution is modified according to:  v 2 is then calculated via:

17. Steffen A. BassHadronization @ RHIC #17 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:

18. Steffen A. BassHadronization @ RHIC #18 Results & Comparison to Data hadron spectra hadron ratios R AA elliptic flow

19. Steffen A. BassHadronization @ RHIC #19 Input and Model Parameters Input for the model is the momentum distributions of constituent quarks and anti-quarks at the time of hadronization the quark distribution is assumed to have a low p t thermal component and a high p t pQCD mini-jet component the thermal component is parameterized as: with a flavor dependent fugacity g a, temperature T, rapidity width Δ and transverse distribution f(ρ,ф) the pQCD component is parameterized as: with parameters C, B and β taken from a lo pQCD calculation

20. Steffen A. BassHadronization @ RHIC #20 Hadron Spectra I

21. Steffen A. BassHadronization @ RHIC #21 Hadron Ratios vs. p t

22. Steffen A. BassHadronization @ RHIC #22 Centrality Dependence of Spectra & Ratios R+F model applicable over full range of centrality deviations from SM as soon as fragmentation sets in low p t deviations due to neglected const. quark mass

23. Steffen A. BassHadronization @ RHIC #23 Flavor Dependence of high-p t Suppression R+F model describes different R AA behavior of protons and pions Lambda’s already exhibit drop into the fragmentation region in the fragmentation region all hadron flavors exhibit jet-quenching

24. Steffen A. BassHadronization @ RHIC #24 Elliptic Flow: Input parton elliptic flow:relative weight of recombination: grey area: region of uncertainty for limiting behavior of R & F hadron v 2 calcuated separately for R and F and superimposed via:

25. Steffen A. BassHadronization @ RHIC #25 Flavor Dependence of Recombination  Recombination describes measured flavor-dependence!

26. Steffen A. BassHadronization @ RHIC #26 Parton Number Scaling of v 2  smoking gun for recombination  measurement of partonic v 2 ! P. Soerensen, UCLA & STAR @ SQM2003 in leading order of v 2, recombination predicts:

27. Steffen A. BassHadronization @ RHIC #27 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 provides a microscopic basis for the Statistical Model issues to be addressed in the future: entropy production treatment of gluons realistic space-time dynamics of parton source need improved data of identified hadrons at high p t

28. Steffen A. BassHadronization @ RHIC #28 The End

29. Steffen A. BassHadronization @ RHIC #29 pQCD approach to parton recombination AA meson double parton scattering scales: single parton scattering and fragmentation scales: T. Ochiai, Prog. Theor. Phys. 75 (1986) 1184

30. Steffen A. BassHadronization @ RHIC #30 Statistical Model vs. Recombination in the Statistical Model, the hadron distribution at freeze-out is given by: usingone obtains: for p t , hadron ratios in SM are identical to those in recombination! (only determined by hadron degeneracy factors & chem. pot.)  recombination provides microscopic basis for SM at large p t see, e.g. Broniowski & Florkowski: PRL 87, 272302 (2001)

31. Steffen A. BassHadronization @ RHIC #31 Hadron Spectra II

32. Steffen A. BassHadronization @ RHIC #32 Elliptic Flow: Recombination vs. Fragmentation high p t : v 2 for all hadrons merge, since v 2 from energy-loss is flavor blind  charged hadron v 2 for high pt shows universal & limiting fragmentation v 2  quark number scaling breaks down in the fragmentation domain