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Quark Coalescence at RHIC

Che-Ming KoTexas A&M University

Introduction Quark coalescence Baryon/meson ratio Hadron elliptic flows and quark number scaling Effect of resonance decays Charm flow Higher-order anisotropic flows Coalescence in transport model Exotica (pentaquark baryon) Entropy problem Summary

Thanks to: Chen, Greco, Levai, Rapp

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Coalescence model in heavy ion collisions

Extensively used for light clusters production First used for describing hadronization of QGP by Budapest group Currently pursued by Oregon: Hwa, Yang (PRC 66 (02) 025205), ……… Duke-Minnesota: Bass, Nonaka, Meuller, Fries (PRL 90 (03) 202303; PRC 68 (03) 044902 ) Ohio and Wayne States: Molnar, Voloshin (PRL 91 (03) 092301; PRC 68 (03) 044901) Texas A&M: Greco, Levai, Rapp, Chen, Ko (PRL (03) 202302; PRC 68 (03) 034904) Most studies are schematic, based on parametrized QGP parton distributions Study based on parton distributions from transport models has been developed by TAMU group (PRL 89 (2002) 152301; PRC 65 (2002) 034904 ) and is now also pursued by D. Molnar (nucl th/0406066)

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Coalescence model

qq3

3

N)p,x(fE)2(

pddp

36/1gg K Mg

)p,...,p;x,...,(xf )p,x(fE)(2

pddpg=N n1n1niiq,i

n

1=i i3

i3

iin ∫∏π

σ

Quark distribution function

Spin-color statistical factor

e.g. 12/1gg *K

Coalescence probabilityfunction

)p,x(fq

PRL 90, 202102 (2003); PRC 68, 034904 (2003)

Number of hadrons with n quarks and/or antiquarks

54/1gg,108/1gg pp

}]/2Δ)m-(m-)p-exp{[(p×

]/2Δ)x-exp[(x=

)p-p;x-x(f=)p,p;x,(xf

2p

221

221

2x

221

212122121M

px

For baryons, Jacobi coordinates for three-body system are used.

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Monte-Carlo method

)p,p;x,x(f

)ppp()j(P)i(Pgpd

dN

jijiM

jTiTT)2(

j,iqqM

T2

M

)p,p,p;x,x,x(f

)pppp()k(P)j(P)i(Pgpd

dN

kjikjiB

kTjTiTT)2(

kjiqqqB

T2

B

Introduce quark probabilities Pq(i) according to theirtransverse momentum and spatial distributions

Allow to treat all quarks on same footing

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5.0≤||,fm5= yτ

• Thermal QGP • Power-law minijets • Choose

GeV2pT GeV2pT

fm8=R

GeV7885.0

y

T

dy

dE

3fm1100⇒ V

Consistent with data (PHENIX)

Parton transverse momentum distributions

149,245 sdu NNN

P. Levai et al., NPA 698 (02) 631

L/

T=170 MeV

quenchedsoft hard

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Other inputs or assumptions

• Minijet fragmentation via KKP fragmentation functions

• Gluons are converted to quark-antiquark pairs with equal probabilities in all flavors.

• Quark-gluon plasma is given a transverse collective flow velocity of β=0.5 c, so partons have an additional velocity v(r)=β(r/R).

• Minijet partons have current quark masses mu,d=10 MeV,

ms=175 MeV, while QGP partons have constituent quark masses

mu,d=300 MeV, ms=475 MeV.

• Use same coalescence radii for all hadrons, i.e., Δx=0.85 fm and

Δp=0.24 GeV.

jet

had

jet2

2jet/had

jet2

had2 p

pz,

z

)Q,z(D

pd

dNdz

pd

dN

7

Au+Au @ 200AGeV (central)

ππρ

Pion and proton spectra

Similar results from other groups

Oregon: parton distributions extracted from pion spectrum

Duke group: no resonances and s+h but use harder parton spectrum

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Baryon/meson ratios

Quark coalescence enhances baryons productionat intermediate transverse momentum

B M

coalescence

fragmentation

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Elliptic flows

Quark v2 extracted from pion and kaon v2 using coalescence model

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Naïve quark coalescence model

Quark transverse momentum distribution

)cos(2)p(v2+1∝)(pf Tq,2Tq φ

Meson elliptic flow

Baryon elliptic flow

)2/p(v2)2/p(v21

)2/p(2v)p(v T2,q

T22,q

T2,qTM2,

)3/p(v3)3/p(v61

)3/p(3v)p(v T2,q

T22,q

T2,qTB2,

Quark number scalingof hadron v2 (except pions):

n)/p(vn

1T2

Only quarks of same momentumcan coalescence, i.e., Δp=0

same for mesons and baryons

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Effects due to wave function and resonance decays

Wave function effectEffect of resonance decays

Wave fun.+ res. decays

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Charm spectra

Charm quark D meson J/ψ

Bands correspond to flow velocities between 0.5 and 0.65

NJ/ψ =2.7 . 10-3

NJ/ψ =0.9 . 10-3

T=0.72 GeVT=0.35-0.50 GeV

13Greco, Rapp, Ko, PLB595 (04) 202

S. Kelly,QM04

Charmed meson elliptic flow

V2 of electrons

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Effect of higher-order parton anisotropic flows

Including 4th order quark flow

)cos(4)p(2v+)cos(2)p(v2+1∝)(pf T4,qTq,2Tq φφ

Meson elliptic flow

Baryon elliptic flow

v

v

3

1+

3

1=

v

v ,

v

v

2

1+

4

1=

v

v ⇒ 2

2,q

4,q2

B2,

B4,22,q

4,q2

M2,

M4,

)v+v(2+1

v+v2= v,

)v+2(v+1

vv2+v2=v 2

4,q22,q

22,q4,q

M4,24,q

22,q

4,q2,q2,q

M2,

)vv+v+6(v+1

v3+vv6+v3+v3= v,

)vv+v+v(6+1

vv6+v3+vv6+v3=v

4,q22,q

24,q

22,q

34,q4,q

22,q

22,q4,q

B4,4,q

22,q

24,q

22,q

24,q2,q

32,q4,q2,q2,q

B2,

Kolb, Chen, Greco, Ko, PRC 69 (2004) 051901

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22,q4,q2

2

4 2v v 1.2 v

v

Data can be described by a multiphase transport (AMPT) model

22,q4,q v v

Data

Parton cascade

Higher-order anisotropic flows

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A multiphase transport model

Lin, PaL, Zhang, Li &Ko, PRC 61, 067901 (00); 64, 041901 (01)

• Initial conditions: HIJING• Parton evolution: ZPC• Hadronization: Lund string model for default AMPT Coalescence model for string melting scenario •Hadronic scattering: ART

• Convert hadrons from string fragmentation into quarks and antiquarks• Evolve quarks and antiquarks in ZPC • When stop interacting, combine nearest quark and antiquark to meson, and nearest three quarks to baryon, • Hadron flavors are determined by quarks’ invariant mass

String melting: PRC 65, 034904 (02); PRL 89, 152301 (02)

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Quark elliptic flows from AMPT

• pT dependence of charm quark v2 is different from that of light quarks• At high pT, charm quark has similar v2 as light quarks• Charm elliptic flow is also sensitive to parton cross sections

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Pseudorapidity dependence of v1 and v2

• String melting describes data near mid-rapidity (||<1.5)• At large rapidity (||>3), hadronic picture works better

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Θ+ production in heavy ion collisions

Quark coalescence model

with quark distribution function and Theta+ Wigner function

Chen, Greco, Ko, Lee and Liu, nucl-th/0308006

4

1

4

1

22224

2

4

2

32/32/3

22

5

5

11513

3

/exp8);(

))2/(exp()(),,(

)21(

)4(

5

1

5

4

);(),()2(

i iiiii

qqq

qs

qsdu

iiiq

i

iii

kypxf

Tmpygpyf

TmVm

mNNNg

ppxxfpxfE

pddpgN

Compare to ~ 0.9 (Randrup, PRC 68) and ~ 0.4 (Letessier et al, PRC 68) from statistical model

Take σ=0.86 fm (RΘ~0.9 fm) gives NΘ~0.64, reduced to 0.19 by relativisticcorrection and sensitive to the Θ+ wave function

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Entropy

For non-relativistic system

thN

Nlog

2

5N

T2

5NS

For g→π assuming mg=mπ

gg

ggg N8.0g

glog-

2

5SS N 5.2S

70% decrease

Coalescence model

4080S 4870S HadQGP 16% decrease

But energy is not conserved ΔE/E~ 18%

Need to take into account binding effect.

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Summary

• Hadronization via parton coalescence seems to work well in understanding observed hadron yields and transverse momentum spectra as well as their elliptic flows.

• Coalescence of minijet partons with partons from QGP

is an alternative mechanism for hadronization of minijet partons.

• Incorporation of parton coalescence in transport models

is useful. For QGP partons, it has already been included in the AMPT model (Lin et al., PRC 65, 034904 (2002); PRL 89, 152301 (2002)). Need to include minijet partons as well.