Recombination

1
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

2
Outline

• Motivation hadron spectra, ratios and flow at
  RHIC
• The recombination idea
• Calculations using recombination fragmentation
• v2 scaling

3
Jet quenching suppression of hard particle
production

• Central AuAucollisions suppression of pions by
  a factor 5
• Suppression of hard (pQCD) hadron production

4
Baryon enhancement at high pt

• For ptgt2 GeV, protons are as abundant as pions
  and kaons!
• hadron production via fragmentation yields p/p
  ratio of 0.1

• where does the large proton over pion ratio at
  high pt come from?
• Why do mesons differ from hadrons?

5
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 ?
  v2?
• novel mechanism of baryon formation?

Sorensen SQM 2003
6

• A possible solution to the puzzle
• parton recombination

Where is pQCD?
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Recombination vs Fragmentation
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, zlt1
recombining partons p1p2ph
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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.

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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

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Recombination formalism I

• Express number of mesons by the quark density
  matrix ?.

• Introduce 2-quark and meson Wigner functions W, ?.

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Recombination formalism II

• choose a hypersurface S for hadronization
• use local light cone coordinates (hadron
  defining the axis)
• wa(r,p) single particle distribution functions
  for quarks at hadronization
• ?M ?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

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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, Cp degeneracy factors)
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Recombination vs. Fragmentation
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
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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.

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• Results Comparison to Data
• hadron spectra
• hadron ratios
• RAA

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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 thermal component and a high pt pQCD mini-jet
  component
• the thermal component is parameterized as
• with a flavor dependent fugacity ga,
  temperature T,
• rapidity width ? and transverse distribution
  f(?,?).
• the pQCD component is parameterized as
• with parameters C, B and ß taken from a lo pQCD
  calculation

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Input and parameters II

• Use hypersurface ? with t2-z2?2 ? 5 fm/c.
• Fix T175 MeV

• Determine
• Radial flow ?0.55 c
• Emission volume
• Energy loss parameter
• Fugacities

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Hadron Spectra I
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Hadron Spectra II
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Hadron Ratios vs. pt
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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
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Elliptic Flow partons at low pt

• azimuthal anisotropy of parton spectra is
  determined by elliptic flow

(?p 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

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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

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• Results Comparison to Data
• elliptic flow

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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

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Flavor Dependence of Recombination

• Recombination describes measured
  flavor-dependence!

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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

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Bill Zajc (DNP Tucson)

• New PHENIX Run-2 result on v2 of ?0s
• New STAR Run-2 result on v2 for ?s
• ALL hadrons measured to date obey quark
  recombination systematics

• smoking gun for recombination
• measurement of partonic v2 !

PHENIX Preliminary
p0
X
STAR Preliminary
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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!

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New developments II

• The ?? will be measured at RHIC. Will v2 scale
  with n5?
• What about other resonances? Influence of the
  hadronic stage?

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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

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The End
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Centrality Dependence of Spectra Ratios

• RF 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

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Flavor Dependence of high-pt Suppression

• RF model describes different RAA behavior of
  protons and pions
• Lambdas already exhibit drop into the
  fragmentation region
• in the fragmentation region all hadron flavors
  exhibit jet-quenching

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Elliptic Flow partons at high pt

• azimuthal anisotropy is driven by parton
  energy/momentum loss ?pt

• L average thickness of the medium

• the unquenched parton pt distribution is shifted
  by ?pt.

• v2 is then calculated via

36
pQCD approach to parton recombination
single parton scattering and fragmentation scales
T. Ochiai, Prog. Theor. Phys. 75 (1986) 1184
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New developments III

• Can we distinguish production scenarios for the
  pentaquark?
• 5q recombination
• KN recombination coalescence,
• KN fragmentation coalescence
• KN fragmentation coalescene in a jet cone (
  5q fragmentation)
• Even obtain information about the structure?