1. Steffen A. BassHadronization @ RHIC #1 Steffen A. Bass Duke University & RIKEN-BNL Research Center The Protons Puzzle 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 in print

2. Steffen A. BassHadronization @ RHIC #2 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

3. Steffen A. BassHadronization @ RHIC #3 Recombination+Fragmentation Model basic assumptions: at low p t, quarks and antiquarks recombine into hadrons locally “at an instant”: hadron momentum P is much larger than average momentum  Δp 2  of the internal quark wave function of the hadron;  features of the parton spectrum are shifted to higher p t in the hadron spectrum parton spectrum has thermal part (quarks) and a power law tail (quarks and gluons) from pQCD.

4. Steffen A. BassHadronization @ RHIC #4 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

5. Steffen A. BassHadronization @ RHIC #5 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

6. Steffen A. BassHadronization @ RHIC #6 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:

7. Steffen A. BassHadronization @ RHIC #7 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

8. Steffen A. BassHadronization @ RHIC #8 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:

9. Steffen A. BassHadronization @ RHIC #9 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

10. Steffen A. BassHadronization @ RHIC #10 Hadron Spectra

11. Steffen A. BassHadronization @ RHIC #11 Hadron Ratios vs. p t

12. Steffen A. BassHadronization @ RHIC #12 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

13. Steffen A. BassHadronization @ RHIC #13 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:

14. Steffen A. BassHadronization @ RHIC #14 Elliptic Flow: Recombination vs. Fragmentation high p t : v 2 for all hadrons merge, since v 2 from energy-loss is flavor blind  quark number scaling breaks down in the fragmentation domain

15. Steffen A. BassHadronization @ RHIC #15 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