Rainer J. FriesRecombination & Fragmentation #1 Rainer J. Fries University of Minnesota Recombination and Fragmentation of Hadrons from a Dense Parton Phase  R.J. Fries, C. Nonaka, B. Müller & S.A. Bass, PRL 90, (2003)  R.J. Fries, C. Nonaka, B. Müller & S.A. Bass, nucl-th/ , JPG t.a.  R.J. Fries, C. Nonaka, B. Müller & S.A. Bass, PRC 68, (2003)  C. Nonaka, R.J. Fries & S.A. Bass, nucl-th/ , submitted to PLB Talk at the RIKEN Workshop on Flow and Collective Phenomena BNL, November 19, 2003

Rainer J. FriesRecombination & Fragmentation #2 Outline Motivation: hadron spectra, ratios and flow at RHIC The recombination idea Calculations using recombination + fragmentation v 2 scaling

Rainer J. FriesRecombination & Fragmentation #3 Jet quenching: suppression of hard particle production Central Au+Aucollisions: suppression of pions by a factor ~5 Suppression of hard (pQCD) hadron production

Rainer J. FriesRecombination & Fragmentation #4 Baryon enhancement at high p t  where does the large proton over pion ratio at high p t come from?  Why do mesons differ from hadrons? For p t >2 GeV, protons are as abundant as pions and kaons! hadron production via fragmentation yields p/π ratio of ~0.1

Rainer J. FriesRecombination & Fragmentation #5 Elliptic flow of K 0 and  hyperon v 2 saturates later and higher than kaon v 2. same effect observed for protons and pions. what drives the different p T scales for K S and Λ v 2 ?  novel mechanism of baryon formation? Sorensen SQM 2003

Rainer J. FriesRecombination & Fragmentation #6 A possible solution to the puzzle:  parton recombination Where is pQCD?

Rainer J. FriesRecombination & Fragmentation #7 Recombination vs Fragmentation 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! recombining partons: p 1 +p 2 =p h fragmenting parton: p h = z p, z<1 Fragmentation:

Rainer J. FriesRecombination & Fragmentation #8 The recombination idea basic assumptions: at low p t, quarks and antiquarks recombine into hadrons on a hadronization hypersurface: hadron momentum P is much larger than masses and momentum scales of the 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 (effective quarks) and a power law tail (quarks and gluons) from pQCD.

Rainer J. FriesRecombination & Fragmentation #9 The nine lives of recombination 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 (field of recent activity!) 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, PRC66, (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/ Z.W. Lin & C.M. Ko, Phys. Rev. Lett 89, (2002) D. Molnar & S. Voloshin, nucl-th/

Rainer J. FriesRecombination & Fragmentation #10 Recombination formalism I Express number of mesons by the quark density matrix . Introduce 2-quark and meson Wigner functions W, .

Rainer J. FriesRecombination & Fragmentation #11 Recombination formalism II choose a hypersurface Σ for hadronization use local light cone coordinates (hadron defining the + axis) w a (r,p): single particle distribution functions 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:

Rainer J. FriesRecombination & Fragmentation #12 Recombination of an exponential spectrum important features: p h = Σ p q d 3 N/dp 3 h  (w q ) n (with n=2,3) for an exponential distribution:  product of all distribution functions only depends on hadron momentum!  results are insensitive to the model used for recombination  Baryon/Meson ratio is independent of momentum, e.g. (C p, C π : degeneracy factors)

Rainer J. FriesRecombination & Fragmentation #13 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:

Rainer J. FriesRecombination & Fragmentation #14 Recombination + Fragmentation Fragmentation of perturbative partons dominates at high p t. Recombination kicks in at 4-6 GeV at RHIC energies. Our description of recombination fails when  /P T and m/P T corrections become large (from 1-2 GeV on at RHIC). But: recombination will still be the dominant hadronization mechanism. Take into account binding energies, mass effects.

Rainer J. FriesRecombination & Fragmentation #15 Results & Comparison to Data hadron spectra hadron ratios R AA

Rainer J. FriesRecombination & Fragmentation #16 Input and Parameters I 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

Rainer J. FriesRecombination & Fragmentation #17 Input and parameters II Use hypersurface  with t 2 -z 2 =  2 ;  = 5 fm/c. Fix T=175 MeV Determine: Radial flow  =0.55 c Emission volume Energy loss parameter Fugacities

Rainer J. FriesRecombination & Fragmentation #18 Hadron Spectra I

Rainer J. FriesRecombination & Fragmentation #19 Hadron Spectra II

Rainer J. FriesRecombination & Fragmentation #20 Hadron Ratios vs. p t

Rainer J. FriesRecombination & Fragmentation #21 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 recombination contribution in spectra

Rainer J. FriesRecombination & Fragmentation #22 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 :

Rainer J. FriesRecombination & Fragmentation #23 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:

Rainer J. FriesRecombination & Fragmentation #24 Results & Comparison to Data elliptic flow

Rainer J. FriesRecombination & Fragmentation #25 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:

Rainer J. FriesRecombination & Fragmentation #26 Flavor Dependence of Recombination  Recombination describes measured flavor-dependence!

Rainer J. FriesRecombination & Fragmentation #27 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

Rainer J. FriesRecombination & Fragmentation #28 Bill Zajc (DNP Tucson) New PHENIX Run-2 result on v2 of  0 ’s: New STAR Run-2 result on v2 for  ’s: ALL hadrons measured to date obey quark recombination systematics PHENIX Preliminary 00  STAR Preliminary  smoking gun for recombination  measurement of partonic v 2 !

Rainer J. FriesRecombination & Fragmentation #29 New developments I Another test: the  meson. Do we see a mass effect or the valence quark structure of hadrons? Reco differs from hydro! The deuteron and the pentaquark should have tremendous v 2. STAR: deuteron v 2 follows the scaling law!

Rainer J. FriesRecombination & Fragmentation #30 New developments II The  + will be measured at RHIC. Will v 2 scale with n=5? What about other resonances? Influence of the hadronic stage?

Rainer J. FriesRecombination & Fragmentation #31 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 One universal parametrization of the parton phase can explain the data! v 2 : proof of collectivity in the parton phase issues to be addressed in the future: entropy & energy resonances and influence of the hadronic phase need improved data of identified hadrons at high p t

Rainer J. FriesRecombination & Fragmentation #32 The End

Rainer J. FriesRecombination & Fragmentation #33 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

Rainer J. FriesRecombination & Fragmentation #34 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

Rainer J. FriesRecombination & Fragmentation #35 Elliptic Flow: partons at high p t azimuthal anisotropy is driven by parton energy/momentum loss Δp t L: average thickness of the medium the unquenched parton p t distribution is shifted by Δpt.  v 2 is then calculated via:

Rainer J. FriesRecombination & Fragmentation #36 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

Rainer J. FriesRecombination & Fragmentation #37 New developments III Can we distinguish production scenarios for the pentaquark? 5q recombination K+N recombination & coalescence, K+N fragmentation & coalescence K+N fragmentation & coalescene in a jet cone (= 5q fragmentation) Even obtain information about the structure?