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Hadronization via Coalescence

Che-Ming KoTexas A&M University

IntroductionQuark coalescence

Baryon/meson ratioHadron elliptic flows and quark number scalingEffect of resonance decaysHigher Fock states Charm flow Higher-order anisotropic flows

Coalescence in transport modelEntropy problem Summary

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Fragmentation leads to p/π ~ 0.2Jet quenching affects bothFragmentation is not the dominant mechanism of hadronization at pT < 4-6GeV

PHENIX, nucl-ex/0212014PHENIX, nucl-ex/0304022

Puzzle: Large proton/meson ratio

π0 suppression: evidence of jet quenching before fragmentation

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Coalescence vs. Fragmentation

Parton spectrum

B M

H

FragmentationLeading parton with pT leads to hadrons of ph=z pT with a probability Dh(z), where z<1

Colascence• partons are already there• ph= n pT ,, n = 2 , 3• Need to be close in phase space• Partonic hydro behavior is shifted to higher pT

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Surprise: quark number scaling of hadron elliptic flow

Except pions, v2,M(pT) ~ 2 v2,q(pT/2) and v2,B(pT) ~ 3 v2,q(pT/3) consistent with hadronization via quark recombination

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Coalescence model in heavy ion collisions

Extensively used for light clusters productionFirst used for describing hadronization of QGP by Budapest groupCurrently 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 parameterized QGP partondistributionsStudy 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 =π

σ⋅∫

36/1gg K ==πMg

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

pddpg=N n1n1niiq,i

n

1=i i3

i3

iin ∫∏ πσ

Quark distribution function

Spin-color statistical factor

e.g. 12/1gg *K ==ρ

Coalescence probabilityfunction

)p,x(f q

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

Number of hadrons with n quarks and/or antiquarks

54/1gg,108/1gg pp ==== ∆∆

]/2∆)m-(m-)p-exp[(p×

]/2∆)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

×

−−δ= ∏

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

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

dN

kjikjiB

kTjTiTT)2(

kjiqqqB

T2

B

×

−−−δ= ∑≠≠

Introduce quark probabilities Pq(i) according to their transverse 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

dydE

3fm1100⇒ ≈V

Consistent with data (PHENIX)

Parton transverse momentum distributions

149,245 ≈≈= sdu NNN

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

L/λ=3.5

T=170 MeV

quenchedsoft hard

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Other inputs and assumptions

Minijet fragmentation via KKP fragmentation functions (Kniehl, Krammer, Potter, NPB 582, 514 (2000))

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 andms=175 MeV, while QGP partons have constituent quark massesmu,d=300 MeV, ms=475 MeV (Non-perturbative effects, Levai & Heinz, PRC 57, 1879 (1998))Use coalescence radii ∆p=0.24 GeV for mesons and 0.36 GeV for baryons

jet

had

jet2

2jet/had

jet2

had2 p

pz,z

)Q,z(Dpd

dNdzpd

dN== ∑∫

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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 ratio

ρ −> ππ

DUKEOREGON

TAMU

12p/π increases by 20% while pbar/π decreases slightly

Baryon/meson ratio at lower energy

Greco, Ko & Vitev, PRC 71, 041901(R) (2005)

r=fraction due to coalescence

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Elliptic flow

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 scaling of hadron v2 (except pions):

n)/p(vn1

T2

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

same for mesons and baryons

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Effects due to wave function and resonance decays

Wave function effect Effect of resonance decays

Wave fun.+ res. decays

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Elliptic flow of resonances

• K* produced during hadronizationhas v2 given by v2,q+v2,s

• K* produced from Kπ scattering has v2 given by

v2,π+v2,K ~ 3v2,q+v2,s

• Observed K* has v2 given by

with r(pT) depending on K* width and Kπ scattering cross section

7.0:)( TPr HG2

QGP2

full2 ))(1()( vPrvPrv TT −+= )( TPr )( TPr

Nonaka et al, PRC 69, 31902 (04)

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Higher Fock States

B. Muller, nucl-th0503003

...321 +++= γγβαβαβα qqqqcgqqcqqcM

Spectra are also not affected(at least for pT >> m )

( )

( )

( ) ( ) ( )2 2

( ) ( ) ( )2 2

v ( ) v /

v ( ) v /

M M M M

B B B B

p c n p n

p c n p n

ν ν νν

ν ν νν

=

=

∑

∑

ν: Fock state, nν = # of partons

Meuller, Fries & Bass, PLB 618, 77 (05)

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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

D

e-

5.2=dydNc

Au+Au @ 200 A GeV

Greco, Rapp & Ko, PLB595 (04) 202

19Greco, Rapp & Ko, PLB595 (04) 202

S. Kelly,QM04

Charmed meson elliptic flow

V2 of electrons

Data consistent with thermalizedcharm quark with same v2 as lightquarks

Smaller charm v2 than light quarkV2 at low pT due to mass effect

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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

vv

31

+31

=vv

,vv

21

+41

=vv

⇒ 22,q

4,q2

B2,

B4,22,q

4,q2

M2,

M4,

)v+v(2+1v+v2

= v,)v+2(v+1

vv2+v2=v 2

4,q22,q

22,q4,q

M4,24,q

22,q

4,q2,q2,qM2,

)vv+v+6(v+1v3+vv6+v3+v3

= v,)vv+v+v(6+1vv6+v3+vv6+v3

=v4,q

22,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 vv ≈⇒≈

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

22,q4,q v v ≈

Data

Parton cascade

Higher-order anisotropic flows

Hydro gives a ratio of ½ (Borghini & Ollitrault, nucl-th/0506045)

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A multiphase transport model

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

• Initial conditions: HIJING (soft strings and hard minijets)• 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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)f'-ff')(fΩ/dd(|v-v|Ωddp∝t)p,(x,f∂p 21212121 ∫ σµµ

2 2

2 2 2 2

d 9 9 1,dt 2(t- ) 2 1 /

s s

sσ πα πασ

µ µ µ≈ =

+

Using αs=0.5 and screening mass µ=gT≈0.6 GeV at T≈0.25 GeV, then <s>1/2≈4.2T≈1 GeV, and pQCD gives σ≈2.5 mb and a transport cross section

σ=6 mb → µ≈0.44 GeV, σt≈2.7 mbσ=10 mb → µ≈0.35 GeV, σt≈3.6 mb

tdd (1 cos ) 1.5mbd

σσ θ≡ Ω − ≈Ω∫

Zhang’s parton cascade (ZPC)

Bin Zhang, Comp. Phys. Comm. 109, 193 (1998)

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Transverse momentum and rapidity distributionfrom default AMPT

BRAHMS Au+Au @ 200 GeV

Default AMPT

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Transverse momentum spectra from AMPT with string melting

Spectra are softer than in default AMPT as current quark massesare used, whose spectra are less affected by collective radial flow

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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

27Need string melting and large parton scattering cross section

Lin & Ko, PRC 65, 034904 (2002)Elliptic flow from AMPT

σp= 6 mb

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Pseudorapidity dependence of v1 and v2

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

Chen, Greco, Ko & Koch, PLB 605, 95 (2005)

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System size dependence of elliptic flowChen & Ko, nucl-th/0505044

Ratio of elliptic flow is ~ 1/3 and scales with the size of colliding systems (~ product of ratios of initial eccentricity (~ ½) and energy density ~2/3)

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Charmed meson elliptic flow from AMPT

Zhang, Chen & Ko, PRC 72, 024906 (05)

Current light quark masses are used in AMPT. With constituent masses will enhance the charmed meson elliptic flow.

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Entropy

For non-relativistic system

−+=

−=

thNNlog

Tm

25N

T25NS µ

For g→π in duality picture with mg=mπ=0 and gluon in equilibrium

gg

ggg N8.0gg

log-25NS N 5.2S ≅

==

ππ 70% decrease

Coalescence model

4080S 4870S HadQGP ≈≈ 16% decrease

But energy is not conserved ∆E/E~ 18%

Need to take into account binding effect such as treated approximately in AMPT by considering invariant mass of coalescence quarks

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Summary

Quark coalescence can explain observedLarge baryon/meson ratio at pT~ 3GeVQuark number scaling of hadron v2

→ signature of deconfinement? Coalescence of minijet partons with thermal partons is significant

→ medium modification of minijet fragmentation. Scaling violation of pion v2 can be explained by resonance decays. Coalescence of thermalized charm quarks can explain preliminary charmed meson spectrum and v2 as well as J/ψ yield. Required quark v2 is consistent with that from parton cascade withlarge parton cross section (~10 mb).

Appreciable parton v4 is seen in parton cascade.Entropy violation (~16%) is not as large as one naively thinks and is related to energy violation of similar magnitude