Hadronization @ RHIC: interplay of fragmentation and recombination

1. Hadronization @ RHIC: interplay of fragmentation and recombination Steffen A. Bass Duke University & RIKEN-BNL Research Center Data: Protons at RHIC - the demise of pQCD? Recombination + Fragmentation Model Results and Predictions • 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 Steffen A. Bass

2. Some of the RHIC Puzzle(s): • protons and the demise of pQCD? • spezies-dependent saturation of v2 Steffen A. Bass

3. The proton puzzle @ RHIC • where does the large proton over pion ratio at high pt come from? • why do protons not exhibit the same suppression as pions? • fragmentation yields Np/Nπ<<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 Steffen A. Bass

4. Elliptic flow of K0 and  • hyperon v2 saturates later and higher than kaon v2. • same effect observed for protons and pions. • the phenomenology seems better described in mT – m0 than pT ; why (kinetic energy)? • what drives the different pT scales for KS and Λ v2? • novel mechanism of baryon formation? Steffen A. Bass

5. A possible solution to the puzzle: • parton recombination Steffen A. Bass

6. Recombination+Fragmentation Model basic assumptions: • at low pt, 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 pt in the hadron spectrum • at high pt, 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 Steffen A. Bass

7. Recombination: Pro’s & Con’s Pro’s: • for exponential parton spectrum, recombination is more effective than fragmentation • baryons are shifted to higher pt than mesons, for same quark distribution • understand behavior of protons! Con’s: fragmenting parton: ph = z p, z<1 recombining partons: p1+p2=ph recombination violates entropy conservation gluons at hadronization need to be converted Steffen A. Bass

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 Steffen A. Bass

9. Recombination: nonrelativistic formalism use thermal quark spectrum given by: w(p) = exp(-p/T) for a Gaussian meson wave function with momentum width ΛM, the meson spectrum is obtained as: similarly for baryons: Steffen A. Bass

10. Recombination: relativistic formalism choose a hypersurface Σ for hadronization use local light cone coordinates (hadron defining the + axis) wa(r,p):single particle Wigner function for quarks at hadronization ФM & ФB: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: Steffen A. Bass

11. results are insensitive to the model used for recombination (light-cone & Wigner functions vs. non-relativistic approx.) • important features: ph = Σ pq d3N/dp3h (wq)n (with n=2,3) • for a thermal distribution: Recombination of thermal quarks • product of all Wigner functions only depends on hadron momentum! • Baryon/Meson ratio is independent of momentum, e.g. (Cp, Cπ : degeneracy factors) Steffen A. Bass

12. Recombination vs. Fragmentation Fragmentation… …nevercompetes with recombination for a thermal (exponential) spectrum: … but it wins out at large pT, when the spectrum is a power law ~ (pT)-b : Steffen A. Bass

13. pt range of parton recombination • quark recombination (coalescence) may dominate for all pt < p0 . Combinatorical models (ALCOR, etc.) work well for total particle yields at SPS and RHIC • low pt is not calculable, but calculation at moderate pt (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” pt evades problems of energy and entropy conservation in recombination: E = (p2+m2)1/2 p Steffen A. Bass

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

15. Elliptic Flow: partons at low pt azimuthal anisotropy of parton spectra is determined by elliptic flow: (Фp: azimuthal angle in p-space) with Blastwave parametrization for parton spectra: azimuthal anisotropy is parameterized in coordinate space and is damped as a function of pt: Steffen A. Bass

16. q r L R Elliptic Flow: partons at high pt azimuthal anisotropy is driven by parton energy/momentum loss Δpt L: average thickness of the medium α(pt): damping function for low pt the unquenched parton pt distribution is modified according to: • v2 is then calculated via: Steffen A. Bass

17. Parton Number Scaling of Elliptic Flow in the recombination regime, meson and baryon v2 can be obtained from the parton v2 in the following way: • neglecting quadratic and cubic terms, one finds a simple scaling law: Steffen A. Bass

18. Results & Comparison to Data • hadron spectra • hadron ratios • RAA • elliptic flow Steffen A. Bass

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 pt thermalcomponent and a high pt pQCD mini-jet component • the thermal component is parameterized as: with a flavor dependent fugacity ga, 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 Steffen A. Bass

20. Hadron Spectra I Steffen A. Bass

21. Hadron Ratios vs. pt Steffen A. Bass

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 pt deviations due to neglected const. quark mass Steffen A. Bass

23. Flavor Dependence of high-pt Suppression R+F model describes different RAA behavior of protons and pions Lambda’s already exhibit drop into the fragmentation region in the fragmentation region all hadron flavors exhibit jet-quenching Steffen A. Bass

24. Elliptic Flow: Input parton elliptic flow: relative weight of recombination: grey area: region of uncertainty for limiting behavior of R & F hadron v2 calcuated separately for R and F and superimposed via: Steffen A. Bass

25. Flavor Dependence of Recombination • Recombination describes measured flavor-dependence! Steffen A. Bass

26. Parton Number Scaling of v2 in leading order of v2, recombination predicts: P. Soerensen, UCLA & STAR @ SQM2003 • smoking gun for recombination • measurement of partonic v2 ! Steffen A. Bass

27. Summary & Outlook The Recombination + Fragmentation Model: • provides a natural solution to the baryon puzzle at RHIC • describes the intermediate and high pt 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 pt Steffen A. Bass

28. The End Steffen A. Bass

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

30. using one obtains: Statistical Model vs. Recombination in the Statistical Model, the hadron distribution at freeze-out is given by: see, e.g. Broniowski & Florkowski: PRL 87, 272302 (2001) • for pt, 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 pt Steffen A. Bass

31. Hadron Spectra II Steffen A. Bass

32. Elliptic Flow: Recombination vs. Fragmentation • high pt: v2 for all hadrons merge, since v2 from energy-loss is flavor blind • charged hadron v2 for high pt shows universal & limiting fragmentation v2 • quark number scaling breaks down in the fragmentation domain Steffen A. Bass