Recombination and fragmentation of hadrons from a dense parton phase
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Recombination and Fragmentation of Hadrons from a Dense Parton Phase. Rainer J. Fries. University of Minnesota. Talk at the RIKEN Workshop on Flow and Collective Phenomena BNL, November 19, 2003. R.J. Fries, C. Nonaka, B. Müller & S.A. Bass, PRL 90, 202303 (2003)

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Recombination and Fragmentation of Hadrons from a Dense Parton Phase

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Recombination and Fragmentation of Hadrons from a Dense Parton Phase

Rainer J. Fries

University of Minnesota

Talk at the RIKEN Workshop on Flow and Collective Phenomena

BNL, November 19, 2003

  • R.J. Fries, C. Nonaka, B. Müller & S.A. Bass, PRL 90, 202303 (2003)

  • R.J. Fries, C. Nonaka, B. Müller & S.A. Bass, nucl-th/0305079, JPG t.a.

  • R.J. Fries, C. Nonaka, B. Müller & S.A. Bass, PRC 68, 044902 (2003)

  • C. Nonaka, R.J. Fries & S.A. Bass, nucl-th/0308051, submitted to PLB

Rainer J. Fries


  • Motivation: hadron spectra, ratios and flow at RHIC

  • The recombination idea

  • Calculations using recombination + fragmentation

  • v2 scaling

Rainer J. Fries

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. Fries

Baryon enhancement at high pt

  • For pt>2 GeV, protons are as abundant as pions and kaons!

  • hadron production via fragmentation yields p/π ratio of ~0.1

  • where does the large proton over pion ratio at high pt come from?

  • Why do mesons differ from hadrons?

Rainer J. Fries

Elliptic flow of K0 and 

  • hyperon v2 saturates later and higher than kaon v2.

  • same effect observed for protons and pions.

  • what drives the different pT scales for KS and Λ v2?

  • novel mechanism of baryon formation?

Sorensen SQM 2003

Rainer J. Fries

A possible solution to the puzzle:

  • parton recombination

Where is pQCD?

Rainer J. Fries

Recombination vs Fragmentation


  • 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!

fragmenting parton:

ph = z p, z<1

recombining partons:


Rainer J. Fries

The recombination idea

basic assumptions:

  • at low pt, 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 pt in the hadron spectrum

  • parton spectrum has thermal part (effective quarks) and a power law tail (quarks and gluons) from pQCD.

Rainer J. Fries

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, 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

Rainer J. Fries

Recombination formalism I

Express number of mesons by the quark density matrix .

Introduce 2-quark and meson Wigner functions W, .

Rainer J. Fries

Recombination formalism II

choose a hypersurface Σ for hadronization

use local light cone coordinates (hadron defining the + axis)

wa(r,p):single particle distribution functions 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:

Rainer J. Fries

  • important features: ph = Σ pq

    d3N/dp3h (wq)n (with n=2,3)

  • for an exponential distribution:

Recombination of an exponential spectrum

  • 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.

(Cp, Cπ : degeneracy factors)

Rainer J. Fries

Recombination vs. 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 :

Rainer J. Fries

Recombination + Fragmentation

  • Fragmentation of perturbative partons dominates at high pt.

  • Recombination kicks in at 4-6 GeV at RHIC energies.

  • Our description of recombination fails when /PT and m/PT 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. Fries

Results & Comparison to Data

  • hadron spectra

  • hadron ratios

  • RAA

Rainer J. Fries

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 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

Rainer J. Fries

Input and parameters II

Use hypersurface  with t2-z2=2;  = 5 fm/c.

Fix T=175 MeV


  • Radial flow =0.55 c

  • Emission volume

  • Energy loss parameter

  • Fugacities

Rainer J. Fries

Hadron Spectra I

Rainer J. Fries

Hadron Spectra II

Rainer J. Fries

Hadron Ratios vs. pt

Rainer J. Fries

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 recombination contribution in spectra

Rainer J. Fries

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:

Rainer J. Fries

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:

Rainer J. Fries

Results & Comparison to Data

  • elliptic flow

Rainer J. Fries

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:

Rainer J. Fries

Flavor Dependence of Recombination

  • Recombination describes measured flavor-dependence!

Rainer J. Fries

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

Rainer J. Fries

Bill Zajc (DNP Tucson)

  • smoking gun for recombination

  • measurement of partonic v2 !

  • 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



STAR Preliminary

Rainer J. Fries

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 v2.

  • STAR: deuteron v2 follows the scaling law!

Rainer J. Fries

New developments II

The + will be measured at RHIC. Will v2 scale with n=5?

What about other resonances? Influence of the hadronic stage?

Rainer J. Fries

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

  • One universal parametrization of the parton phase can explain the data!

  • v2: 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 pt

Rainer J. Fries

The End

Rainer J. Fries

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

Rainer J. Fries

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

Rainer J. Fries

Elliptic Flow: partons at high pt

azimuthal anisotropy is driven by parton energy/momentum loss Δpt

L: average thickness of the medium

the unquenched parton pt distribution is shifted by Δpt.

  • v2 is then calculated via:

Rainer J. Fries

double parton scattering scales:




pQCD approach to parton recombination

single parton scattering and fragmentation scales:

T. Ochiai, Prog. Theor. Phys. 75 (1986) 1184

Rainer J. Fries

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?

Rainer J. Fries

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