Recombination of Quarks (and Statistical Model)

1
Recombination of Quarks (and Statistical Model)

• Rainer Fries
• Texas AM University RIKEN BNL

Workshop on Hadronization ECT, Trento, September
3, 2008
2
Overview

• Hadronization Introduction
• RHIC Results
• Recombination Models
• Dilute Systems
• Conclusions

3
Thoughts on Hadronization
4
Hadronization

• Formulation of the Problem
• How does an ensemble of partons C turn into an
  ensemble of hadrons H (w/ or w/o being coupled to
  a medium or spectators).
• Solving this problem is rather hopeless.
• Less ambitious look for special cases in which
  some non-trivial statements can be made.

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Hadronization

• Can the parton ensemble C be well-defined?
• It is an intermediate state in a quantum field
  theory!
• C and Y are unobserved in a scattering reaction
  AB ? HX
• We have to sum over all possible pairs (C,Y)
• Interference between (C,Y) and (C,Y) in
  amplitude and complex conjugated amplitude.
• Probably it is not always possible to have well
  defined C with probabilistic interpretation!
• But for all successful descriptions of
  hadronization this is the case.

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Example Fragmentation

• Single inclusive production of a single hadron h
  at large momentum (e.g. in ee-, pp)
• QCD factorization process dominated by single
  parton in the intermediate state
• No statement about hadronization itself.
• But now one can write down a well-defined
  operator definition for the process with a
  probabilistic interpretation a fragmentation
  function
• These matrix elements can be measured and have
  certain universality properties. They are very
  hard to calculate.

Collins and Soper many others since
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Example Hard Exclusive Processes

• Hard process in which nucleon stays intact, e.g.
  form factor in p ? ? p

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Example Hard Exclusive Processes

• Hard process in which nucleon stays intact, e.g.
  form factor in p ? ? p
• Here parton ensemble C complete
  set of valence quarks of the nucleon
• Sensitive to matrix element
• ? nucleon light cone wave function
• ? describes amplitude uud ? p
• resembles recombination!

Chernyak and Zhitnitsky Brodsky and Lepage
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Example Hard Exclusive Processes

• Some constraints on ? from
  symmetries, some experimental
  constraints at least for
  pions.
• In terms of light cone fraction x
• Caveat
• The way a hadron looks depends
  on what and
  how we measure.
  
• Hard exclusive LC wave functions are
  special perturbative scale,
  infinite
  momentum frame,
  exclusive
  process.

E791 ?A?2J
How we see a hadron depends on
which process we use to probe
the resolution of the process
the reference frame.
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Statistical Model

• Statistical hadronization avoids the question of
  an explicit initial parton state.
• hadrons born into equilibrium
• But it invites the thought that some kind of
  statistical description should also be available
  for the partonic side of the process.
• SHM certainly not applicable to exclusive
  processes

SHM lives here
How can it be connected to this?
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Hints from RHIC
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Hot Nuclear Matter

• Early universe
• A hot soup of quarks gluons
• Phase transition to hadrons
• Temperature T 1012 K
• Lattice QCD
• Phase transition or cross over at Tc 170 192
  MeV (_at_ ?B 0)
• ? critical point ?
• Strongly interacting partons or weakly
  interacting gas above Tc?

Karsch et al.
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Jet Quenching

• RHIC strong quenching of high-PT pions and
  kaons.
• Energy loss of leading parton.
• Naïve pre-2002 expectation hadron production
  from jets above PT 2 GeV.

Nuclear modification factor
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Jet Quenching Baryon Puzzle

• RHIC strong quenching of high-PT pions and
  kaons.
• Energy loss of leading parton.
• No jet quenching for baryons? (RAA , RCP 1)
• Seen for PT 1.5 5
  GeV/c.
• Baryon Anomaly at
  intermediate PT.

PHENIX
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Baryon Puzzle

• Proton/pion ratio gt 1 at PT 4 GeV in AuAu
  collisions.
• Expectation from parton fragmentation p/? 0.1
  0.3
• As measured in pp and ee-

PHENIX
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Baryon vs Meson

• General baryon/meson pattern p, ?, ?, ? versus
  K, ?, ?, K, ?

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Elliptic Flow v2

• Azimuthal anisotropy for finite impact
  parameter b gt 0
• 3 mechanisms to translate spatial
  anisotropy in the initial state into
  
  momentum anisotropy in the final state.

Turbide, Gale, RJF
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Elliptic Flow Scaling

• Scaling first found experimentally
• n number of valence quarks
• Very much unlike hydrodynamics (mass ordering)

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Elliptic Flow Scaling

• Low PT scaling with kinetic energy
• Implied by hydrodynamics.
• Scaling close to perfect.

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Recombination Models at RHIC were born out of an
apparent failure of jet fragmentation at
intermediate PT
Hydrodynamics didnt save the day.
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Why not Hydro?

• Baryon vs meson doesnt seem to be compatible
  with either hydro nor the statistical model.
• No mass effect ? behaves like a pion (m? ? mp,
  m? gtgt m?)

STAR
STAR
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Recombination Models
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Dense Parton Systems

• Basic idea
• Fragmentation limit of hadronization for very
  dilute systems (parton density ? 0)
• Opposite limit thermalized phase of partons
  just above Tc
• No perturbative scale in the problem (T ? ?QCD)
• Naively recombine partons that
  are already filling phase
  space.

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Instantaneous Coalescence

• Simple realization of a recombination model
• Recombine valence quarks of hadrons
• Dressed quarks, no gluons
• Instantaneous projection of quark states (density
  matrix ?) on hadronic states with momentum P
• Effectively 2 ? 1, 3 ? 1 processes
• Projection conserves only 3 components of
  4-momentum.

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Instantaneous Coalescence
Meson Wigner function
Production hypersurface

• Hadron spectra can be
  written as convolution of
  
  Wigner functions W, ?
• Replace Wigner function by
  classical phase space
  distribution
• MC implementation available
• Collinear approximation
• light cone formalism, PT gtgt M

Quark Wigner function
Greco, Ko Levai
RJF, Müller, Nonaka Bass Hwa Yang also
Rapp Shuryak
Can be modeled with hard exclusive light
cone wave functions
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Transport Approach

• Boltzmann approach applied to ensemble of quarks
  and antiquarks scattering through meson
  resonances.
• Breit-Wigner cross sections
• Properties
• Conserves energy and momentum.
• Finite time to reach equilibrium.
• First studies good description of spectra and
  elliptic flow compatible with KET scaling.
• Baryons difficult.

Ravagli Rapp Ravagli, van Hees Rapp
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The Thermal Case

• Thermal parton spectra yield thermal hadron
  spectra.
• For instant. recombination (collinear case)
• Should also hold in the transport approach.
• Automatically delivers NB NM if mass effects
  are suppressed
• Details of hadron structure are not relevant for
  thermal recombination at high momentum (collinear
  case).
• Wave function can be integrated out.
• Important for elliptic flow scaling.
• Also seen numerically in full 6-D phase space
  coalescence.

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Exp Power Laws

• Comparison of different scenarios
• Power law parton spectrum
• Recombination is suppressed
• Good QCD factorization should
  hold at least for
  asymptotically
  large momentum
• Exponential parton spectrum
• Recombination more effective
• Even larger effect for baryons

fragmenting parton ph z p, zlt1
recombining partons p1p2ph
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Phenomenology

• Dual model of hadron production
• Recombination pQCD/fragmentation.
• Using thermal quark spectra for recombination.
• T 175 MeV
• Radial flow ? 0.55
• Fit to pion data ? predictive power for all other
  hadron species
• Describes hadron production at RHIC in the
  collinear region (for PT gt 12 GeV/c).

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Phenomenological Success

• Recombination of thermal partons dominates up to
  4 GeV/c for mesons, 6 GeV/c for baryons

Greco, Ko Levai
RJF, Müller, Nonaka Bass
RJF, Müller, Nonaka Bass
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Particle Ratios Statistical Model

• Comparison of ratios to statistical model
• SM is formally recovered by the instant.
  recombination model in the limit P ? ? for
  thermal quark spectra.
• In reality
• Deviations at low PT due to mass effects.
• Jet physics takes over at around 4-6 GeV/c.
• Modern data

STAR
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Elliptic Flow Scaling

• Assume universal elliptic flow v2p of the partons
  before the phase transition
• Recombination prediction
• Factorization of momentum
  and position space
• Recover scaling law for infinitely narrow wave
  functions
• Scaling holds numerically also for less special
  choices.

Momentum shared fractions x and 1-x
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Elliptic Flow Scaling

• Data Compilation

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Elliptic Flow in Transport Approach

• Kinetic energy scaling preserved going from the
  quark to the hadron phase.
• Test with light and heavy quarks
• Comparison with baryons?

Ravagli and Rapp
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Dilute Parton Systems
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Recombination in Other Systems

• Recombination at very forward rapidity
• No hard scale
• Recombine beam remanants/spectators.
• Leading Particle Effect (forward rapidities)
• D/D? asymmetries clearly not described
  by pQCD
  fragmentation
• Explained by recombination with beam
  remnants

K.P. Das R.C. Hwa Phys. Lett. B68, 459
(1977) Quark-Antiquark Recombination in the
Fragmentation Region
E791 ?? beam
E791 ?- beam hard cc production recombine c
with d valence quark from ?- gt reco of c with d
Braaten, Jia Mehen
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Parton Shower Recombination

• Attempts to treat reco fragmentation
  consistently
• jets ? parton showers fitted to fragmentation
  functions
• Also 2- and 3- quark constituent quark
  fragmentation recombination (? Q2 evolution)
• HY Model recombine all partons
• Partons soft/thermal showers from jets
• 2-parton distribution function

Hwa, Yang
Majumdar et al.
Partons from 1 jets
soft-soft
Partons from 2 jets
soft-shower
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HY Parton Showers

• Shower parton distribution in the HY Model
• Hard parton i
  shower parton j, momentum zpi
• Determined by fits to fragmentation functions.
• Could they resemble thermal boost ??

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The Bigger Picture

• Thermal recombination fragmentation treated
  equally
• Can calculate cross terms.
• Applicable to all kind of scattering systems.
• Assume some exponential bulk (not thermal!) in
  pp or pA to account for soft physics.
• Usually just adds two parameters.
• E.g. check for baryon enhancement.

ee-
pp
pA
AA
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Application to pA

• dAu / pA at midrapidity
• Cronin enhancement initial state broadening
• At RHIC large final state effect seen baryon
  enhancement.
• Pick-up reactions (soft/hard recombination)
  important

Hwa, Yang
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Conclusions
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Summary

• Thermal medium probably ok
• Jets (vacuum)
• But there is a continuum of possibilities between
  those.

Stat. Model
Recombination
Stat. Model
Fragmentation
Shower Recombination Clusters
?
(?)
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Just In

• The Ridge and the Peak

?

• Baryon/meson ratios
• jet smaller than inclusive
• and similar to pp
• ridge similar to inclusive

?
STAR
pTtrig gt 4.0 GeV/c 2.0 lt pTassoc lt pTtrig
C. Suarez (STAR), poster, QM2008
AuAu 2ltpTtriglt3 GeV/c,CuCu3ltpTtriglt6 GeV/c
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Backup
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Quark Counting Rule for the QGP

• Quark counting rules quark substructure in
  hadrons
• Classic example counting valence quarks
• RHIC a new quark counting rule
• Subhadronic degrees of freedom are at work.
• They act collectively observable v2 describes
  collective effect
• Equilibrium / hydrodynamic behavior of this
  matter (?)
• Deconfinement is reached.

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Hadron Correlations

• Jet-like correlations seen even at intermediate
  PT.
• How to reconcile with thermal recombination?
• Correlations induced by Soft/Hard Reco (pick up
  reactions)
• Hadron correlations arise from correlations
  between soft partons

Hot spots fully or partially thermalized jets ?
correlated soft partons
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Hadron Correlations

• Jet like correlations seen in data even at
  intermediate PT.
• How to reconcile with recombination?
• Correlations induced by Soft/Hard Reco (pick-up
  reactions)
• Hadron correlations arise from correlations
  between soft partons
• Model with 2-body correlations

Preliminary RJF Nonaka frag-frag reco-reco
only
Near side
Away side
central
Meson trigger
Baryon trigger
peripheral
RJF, Bass Müller
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LHC Preview

• Recombination window might widen at LHC.
• It depends on the interplay of increased redial
  flow and energy loss.

RJF Müller