& RIKEN-BNL Research Center Dynamics of Hadronization: Interplay of Fragmentation and Recombination Steffen A. Bass Duke University & RIKEN-BNL Research Center The baryon puzzle at RHIC Recombination + Fragmentation Model Results: single particle observables, correlations & entropy 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 Steffen A. Bass

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

The baryon 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

Species dependent saturation of elliptic flow hyperon v2 saturates later and higher than kaon v2. same effect observed for protons and pions. at low pT the phenomenology seems better described in mT – m0 than pT , indicating hydro scaling, yet scaling breaks down for high pT what drives the different pT scales for KS and Λ v2? novel mechanism of baryon formation? Steffen A. Bass

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

Recombination: Pro’s & Con’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 simple recombination may violate entropy conservation gluons at hadronization need to be converted Steffen A. Bass

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) Steffen A. Bass

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

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

Recombination vs. Fragmentation … never competes 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

Hadron Spectra Steffen A. Bass

Hadron Ratios vs. pt Steffen A. Bass

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

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

Parton Number Scaling of v2 in leading order of v2, recombination predicts: smoking gun for recombination measurement of partonic v2 ! Steffen A. Bass

Resonance v2: scaling violations QGP resonances: hadronizing QGP, no rescattering HG resonances: hadronic phase, h-h rescattering quarks n=2 scaling n=4 scaling Key: v2 is additive for composite particles Total is determined by experiments and related to width of particles and cross section in the hadronic medium. Steffen A. Bass

Two-Particle Correlations: a Challenge? PHENIX & STAR measure associated yields in pT 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 pT very similar to p+p; jet fragmentation? analyze as function of integrated yield: simple recombination of uncorrelated thermal quarks cannot reproduce two particle correlations Steffen A. Bass

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: Steffen A. Bass

Correlations: Proof of Principle Meson-trigger Baryon-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 Steffen A. Bass

Recombination: Entropy Puzzle quark recombination hadronization 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 Steffen A. Bass

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

Entropy in the Deconfined Phase Lattice QCD [CP-PACS with Nt=6 & Nf=2] entropy content of the deconfined phase near TC is strongly reduced due to interactions! systematic uncertainties include: thermodynamic limit, continuum limit, unphysically large quark mass… Steffen A. Bass

Entropy Puzzle resolved? Quark Phase Hadron Phase Recombination may be compatible with the entropy constraint after all! ? SH is larger than often assumed. SQ near TC is strongly reduced due to interactions. No direct comparison of the entropy content of both phases: Volume at hadronization ? Number of quarks on the lattice ? Steffen A. Bass

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 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 pt Steffen A. Bass

The End Steffen A. Bass

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

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