Kang Seog LeeChonnam National University, Korea

Dynamical Recombination model of QGP

Dynamical Recombination model of QGP

• Introduction – recombination model• Dynamic recomination calculation : Hydrodynamic evolution of QGP + + Hadronization via recombination + Hadronic rescattering by URQMD • Results• Summary and Outlook collaborators

:• S. Bass • B. Müller• C. Nonaka

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Recombination : long historyRecombination : long historyHigh 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• R. Fries, B. Mueller, C. Nonaka & S.A. Bass, Phys. Rev. C68, 044902 (2003)• V. Greco, C.M. Ko and P. Levai, Phys. Rev. Lett. 90 - AMPT

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+Fragmentation Model

Recombination+Fragmentation Model

basic assumptions:• at low pt, the quarks and antiquark spectrum is

thermal and they recombine into hadrons locally “at an instant”:

• 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

qq M qqq B

R.J. Fries, C. Nonaka, B. Mueller & S.A. Bass, PRL 90 202303 (2003) ; PRC68, 044902 (2003)

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Hadron Spectra IHadron Spectra I

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Elliptic Flow: Recombination vs. Fragmentation

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 domainkslee

Parton Number Scaling of v2

Parton Number Scaling of v2

smoking gun for recombination

measurement of partonic v2 !

P. Soerensen, UCLA & STAR @ SQM2003

2 2

2 2

22

33

pM

p

t

B t

t

t

pv

p

v

p v

p

v

•in leading order of v2, recombination predicts:

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30-Mar-09 W.A. Zajc

Parton Number Scaling of v2Parton Number Scaling of v2

Hadron Ratios vs. ptHadron Ratios vs. pt

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Energy systematics – 200 and 62• Mid-rapidity 62 GeV similar to ~2.2 at 200

GeV with significant P/ ratios• At y~0 and y~2.2 significant medium effects• Close to beam rapidity (forward at 62 GeV)

P/ exhibits no centrality dependence and is similar to pp indicating little influence from media ( quark coalescence, hadronic re-scattering)

QM 2009, Knoxville9•Talk by Pawel Statzel, Friday

Exploring QCD Matter at RHIC and LHC

Exploring QCD Matter at RHIC and LHC

initial state

pre-equilibrium

QGP andhydrodynamic expansion

hadronization

hadronic phaseand freeze-out

QGP • hydrodynamic evolution - C. Nonaka• reasonable for a perfect fluid

Hadronization via recombination

• recombination

Hadronic rescattering

• URQMD - S. Bass

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Recombination of thermal quarks IRecombination of thermal quarks I

3 3

i1 2 1 23 3

( , )(2 )

( , )(2 )

MMi i

ab

BBi

abc

dN dE G k P k

d P

ddNE G k k P k k

d

P d

pP d

3

3 32

2

M

2

B 1 2 121 2 1 21

( )

(

( ) ( )

( ) ( ) ( ) , , )

a b

a b c

Mab

Babc

w k w P k

w k w k w P k

k

k k P k k

G

k k

d k

G d k d

• wave functions are best know in the light cone frame. • for a thermal distribution:

( , ) exp( / )qw r k k u T

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Recombination of thermal quarks IIRecombination of thermal quarks II

For a cell at hadronization, is the energy available for hadronization where is the volume fraction of QGP phase.

• hadronization hypersurface : mixed phase• Energy conservation during the recombination

iE

/Q totV V

• In a hydrodynamic cell, i, hadronization rate in direction is proportional to i

i

p d p d

1i i i

cell i hadi j

i j

E E

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Recombination of thermal quarks IIIRecombination of thermal quarks III

Inside a group, relative abundances are assumed to be

( )// /i jm m Ti j i jN N e C C (Ci : degeneracy factors)

1, 2, 3, 4qq M qs M sq M ss M

• Since , there are 4 cases for mesons and 8 cases for baryons.

q sm m

1, 2, 3, 4qqq B qqs B qss B sss B

• depends only on the quark masses and is independent of hadron mass.

1 2( , )MabG k k

5, 6, 7, 8qqq B qqs B qss B sss B

e.g. 1 ( , , ,...)M kslee

Recombination of thermal quarks IVRecombination of thermal quarks IV

• After hadronization, hadronic rescattering is simulated by URQMD !

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Results • hadron spectra• hadron ratios – coming soon

• RAA – coming soon

• elliptic flow – coming soon•Comparison with statistical model

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Summary & Outlook

Summary & Outlook

Hadronization of a QGP via Recombination:• Hydrodynamic evolution of a QGP until mixed phase - reasonable for a perfect fluid• hadronization via quark recombination - quark number scaling of elliptic flow, …• hadronic rescattering by URQMD - different chemical and thermal freeze-out No problem of freeze-out prescription !!!issues to be addressed in the future:• results are coming soon.• comparison with statistical hadronization• combine with viscous hydrodynamics or parton cascade

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The End

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p

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K

p

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K

p

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Statistical Model vs. Recombination

Statistical Model vs. Recombination

3

3

3

1

exp / 12with

i

ii

iB s I

i

i i

d NE P d

df P u

gf P u

P u B I T

P

S

•in the Statistical Model, the hadron distribution at freeze-out is given by:

• If , recombination is identical to SM!

3

3 32

2

M

2

B 1 2 121 2 1 21

( )

(

( ) ( )

( ) ( ) ( ) , , )

a b

a b c

Mab

Babc

w k w P k

w k w k w P k

k

k k P k k

G

k k

d k

G d k d

( ) ( ) ( )h a bE P E k E P k

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Hadron Spectra IIHadron Spectra II

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Elliptic Flow: partons at low pt

Elliptic Flow: partons at low pt

2

2 11 2 cos 2

2 pt t p t t

d N dN

p dp d p dv

p

2

2 10

22

0 10

sinh coshcos 2

cos 2sinh cosh

st ts s

t p

t ts

s

s s

p md I K

T Tv p

p md I K

T T

0 2

0

111ln 1 cos 2

2 1and

1 /p t p

ts t

t

st

p pp p

• azimuthal anisotropy of parton spectra is determined by elliptic flow:

• with Blastwave parametrization for parton spectra:

(Фp: azimuthal angle in p-space)

• azimuthal anisotropy is parameterized in coordinate space and is damped as a function of pt:

Recombination: relativistic formalism

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:

EdN

M

d 3P d

P u(2 )3

dx w (R, xP ) ,

w (R,(1 x)P )

M(x)

2

EdN

B

d 3 p d

P u(2 )3

dx dx ' w (R, xP ) , ,

w (R, x ' P ) w (R,(1 x x ')P )

B(x, x ')

2

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