Kang Seog Lee

Chonnam National University

Hadronization of a QGP via recombination

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

collaborators:• S. Bass • B. Müller• C. Nonaka

Recombination : 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

Anisotropic flow:• S. Voloshin, QM2002, nucl-ex/020014• Z.W. Lin & C.M. Ko, Phys. Rev. Lett 89, 202302 (2002) : AMPT• D. Molnar & S. Voloshin, nucl-th/0302014 Jin-Hui Chen’s

talk

Recombination+Fragmentation Model

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

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

features of the parton spectrum are shifted to higher pt in the hadron spectrum

• 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)

Hadron Spectra I

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

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:

Hadron Ratios vs. pt

Model

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

Recombination 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

Recombination 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

Recombination 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

Recombination of thermal quarks IV

• After hadronization, hadronic rescattering is simulated by URQMD !

Results & Comparison to Data• hadron spectra• hadron ratios – coming soon

• RAA – coming soon

• elliptic flow – coming soon

K

p

K

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

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

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:

3

2

2

2

2

2 2

2

2

2

3 322

1

an

2

d

3

12

3

63

p pt tp t

Mt

p pt t

Bt

p pp v vv

p pv

p pv v

v

neglecting quadratic and cubic terms, one finds a simple scaling law:

2 2 2 2an22 3

d 3p pM tt

B tt

pv p vv

pv p

p

p

p

Recombination: Pro’s & Con’s

Pro’s:• 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!Con’s:

• recombination violates entropy conservation• gluons at hadronization need to be converted

recombining partons:p1+p2=ph

fragmenting parton:ph = z p, z<1