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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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