LBNL School on

Twenty Years of Collective Expansion

Berkeley, 19-27 May 2005

Berndt Müller

Duke University

Re-Combinatorics of Thermal Quarks

• M. Asakawa• S.A. Bass• R.J. Fries• C. Nonaka

• PRL 90, 202303 • PRC 68, 044902• PLB 583, 73• PRC 69, 031902• PRL 94, 122301• PLB (in print)

Special thanks to: The Duke QCD theory group

Jet quenching seen in Au+Au, not in d+Au

PHENIX Data: Identified 0

d+Au

Au+Au

Suppression Pattern: Baryons vs. Mesons

What makes baryons different from mesons ?

…or what really came as a complete surprise…

Suppression: Baryons vs. mesons

baryons mesons

CP

pT (GeV/c)

behaves like meson ?

(also -meson)

Hadronization Mechanisms

qq

q

Baryon1

Meson

Fragmentation

q q

q q q

Baryon1

Meson

Recombination

M Q B Q2 3p p p p

Recombination was predicted in the 1980’s – Hwa, Ochiai, …

S. Voloshin

QM2002

Recombination is favored …

… for a thermal source

Baryons compete with mesons

Fragmentation wins out for a power law tail

Instead of a History ….

• Recombination as explanation for the “leading particle effect”:– K.P. Das & R.C. Hwa: Phys. Lett. B68, 459 (1977)

– Braaten, Jia, Mehen: Phys. Rev. Lett. 89, 122002 (2002)

• Fragmentation as recombination of fragmented partons:– R.C. Hwa, C.B. Yang, Phys. Rev. 024904 + 024905 (2004)

• Relativistic coalescence model– C.B. Dover, U.W. Heinz, E. Schnedermann, J. Zimanyi,

Phys. Rev. C44, 1636 (1991)

• Statistical recombination– ALCOR model (see T.S. Biro’s lecture)

• Quark recombination / coalescence– Greco, Ko, Levai, Chen, Rapp / Lin, Molnar / Duke group

– A. Majumder, E. Wang & X.N. Wang (in progress)

Recombination: The Concept

“fragmentation”

“recombination”REC

DER

1 2p p P

“freeze-out”

Sudden recombination picture

q

qM

d

Transition time from QGP into vacuum (in rest frame of produced hadron) is:

/fT

md d

p

Not gradual coalescence from dilute system !!!

pT m

Allows to ignore complex dynamics in hadronizationregion; corrections O(m/pT)2

QGP

Tutorial: Non-Relativistic Recombination

Consider system of quarks and antiquarks (no gluons!) of volume V and phase-space distribution wa(p) = p,a p,a .

Quark-antiquark state vs. meson state:

1 1 2 2( )11 2

1

,

, ( )

i p x p x

iP RM

x p p V e

x P V r

Q

M e

11 22

1 2

( )

)

R x x

r x xwith

Probability for finding a meson with P and q = (p1-p2)/2 :3

2 231 2 1 22

(2 )ˆ, , ( ) ( )MQ p p M P P p p q

VNumber of produced mesons:

3 2

3

3 3323 1 2

1 1 1 23 3 3

, ,(2 )

( ) ( ) , ,(2 ) (2 ) (2 )

M

abM a b

ab

d PN V M P M P

d p d pd PC V w p w p Q p p M P

Tutorial – page 2

321 1

2 23 3 3ˆ ( )

(2 ) (2 )abMM a b M

ab

dN V d qC w P q w P q q

d P

The meson spectrum is given by:

Consider case P q, where q is of order M, and expand (for wa = wb):

21 1 1 1 1 1 12 2 2 2 2 2 2i j i j i j

ij

w P q w P q w P q q w P w P w P w P

Using only lowest order term:

32 221 1

2 23 3 3 3ˆ ( )

(2 ) (2 ) (2 )M

M M M

dN V d q VC w P q C w P

d P

Corrections are of order M2 w/Pw = M

2/PT for thermal quarks.

Tutorial – page 3

The same for baryons:3

2 231 2 3 1 2 32

(2 )ˆ, , ( ) ( , )BQ p p p B P P p p p q s

V

where q, s are conjugate to internal coordinates

11 2 3 1 22 ( ) a d 'nr x x x r x x

The baryon spectrum is then:

3 321 1 1 1 1

3 2 3 2 33 3 3 3

3 2133

ˆ ( , )(2 ) (2 ) (2 )

1 ( / )(2 )

abcBM a b c B

abc

B B

dN V d q d sC w P q s w P q s w P q q s

d P

VC w P O PT

Tutorial - page 4

For a thermal Boltzmann distribution ( ) exp ( ) /w p E p T

we get

21 12 2

31 13 3

exp 2 / exp /

exp 3 / exp /

M

B

w P E P T E T

w P E P T E T

and therefore: 2( )1 ( / )

( )B B

M M

dN E CO PT

dN E C

Wigner function formulation

General formulation relies on Wigner functions:

1 2 2

3 31 21 1 1 1

1 1 2 2 1 1 2 2 1 2 1 22 2 2 2 3 3, , ( , ; , )(2 ) (2 )

p r p rab

d p d pr r r r r r r r e W r r p p

Meson number becomes:3 3

3 3 3 1 1 1 12 2 2 23 3

*1 12 2

, ; ,(2 ) (2 )M ab

ab

iq rM M

d P d qN d Rd r d r W R r R r P q P q

e r r r r

Relativistic generalization (u = time-like normal of volume):

3 3 3P ud P d R d P d

E

Relativistic formulation

2M

M3 3,

2B

B3 3, ,

( , ) ( , (1 ) ) ( )(2 )

' ( , ) ( , ' ) ( , (1 ') ) ( , ')(2 )

dN P uE d dxw R xP w R x P x

d P

dN P uE d dxdx w R xP w R x P w R x x P x x

d p

Relativistic formulation using hadron light-cone frame (P = P ):

For a thermal distribution,

the hadron wavefunctions can be integrated out, eliminating the model dependence of predictions. This is true even if higher Fock space states are included!

( , ) exp( v / )w r p p T

03 2 01

2with and

kd k dk d k k k k k xP

k

Beyond the lowest Fock state1

2 21

0

12

2

0

; ( 1) ( , ; ) ( ) ( )

( 1) ( , , ; ) ( ) ( ) ( )

a b a b a b a b

a b c a b c a b c a b c

M Q dx dx x x x x Q q x q x

dx dx dx x x x x x x Q q x q x g x

12

2

0

12/

2

0

( ) ( ) (( 1) ( , , )

( , ,1 )

)q a qqqg a b c a b c a b c

P Ta b a b a b

a g cW dx dx dx x x x x x x

e

w x w x w

dx dx x x x x

x

/ / / ( ) /a b c a b cx P T x P T x P T x x x P Te e e e

1 12 2/

1 2

0

/

0

( ,1 ) ( , ,1 )P Tqq qqg a a

P Tb a b a a bM bW W e dx x x dx dx x x x xW e

12

1

0

1 12 2/

1 1

0 0

( 1) ( , )

( )

( ) ( ) ( ) ( )

( 1) ( , ) ( ,( ) 1 )

qq a b a b a b a b a b

P Ta b q a q aa b a b a a aw x w x

W dx dx x x x x q x q x q x q x

dx dx x x x x e dx x x

/ / ( ) /a b a bx P T x P T x x P Te e e For thermal medium

Statistical model vs. recombination

3

1

3

3 exp / 12

with

B s

i

ii i i iI

id NE P d

df P u

gf P u P u B S I T

P

In the stat. model, the hadron distribution at freeze-out is given by:

For pt , hadron ratios in SM are identical to those in recombination!

(only determined by hadron degeneracy factors & chem. pot.)recombination provides microscopic basis for apparent chemical

equilibrium among hadrons at large pt

BUT: Elliptic flow pattern is approximately additive in valence quarks, reflecting partonic, rather than hadronic origin of flow.

Recombination vs. Fragmentation

1h 1

h3 3 30

( , ) ( )(2 ) z

dN P u dzE d w r P D z

d P zFragmentation:

…always wins over fragmentation for an exponential spectrum (z<1):

… but loses at large pT, where the spectrum is a power law ~ (pT) -b

ee xxp p(v / )) v /( PP zTT

Recombination… ( , ) ( , (1 ) ) exp v /

( , ) ( , ' ) ( , (1 ') ) exp v /

w r xP w r x P P T

w r xP w r x P w r x x P P T

Meson

Baryon

2M

M3 3,

( , ) ( , (1 ) ) ( )(2 )

dN P uE d dxw R xP w R x P x

d PRecombination:

Model fit to hadron spectrum

Teff = 350 MeVblue-shifted temperature

pQCD spectrum shifted by 2.2 GeV

R.J. Fries, BM, C. Nonaka, S.A. Bass (PRL )

Corresponds to = 0.6 !!!

Recall:

G 0.5 ln(…)

Q 0.25 ln(…)

Hadron Spectra I For more details – see S.A. Bass’ talk tomorrow

Hadron Spectra II

Hadron dependence of high-pt suppression

• R+F model describes different RAA behavior of protons and pions

• Jet-quenching becomes universal in the fragmentation region

Hadron production at the LHC

r = 0.75

r = 0.85

r = 0,65

R.J. Fries

includes partonenergy loss

Conclusions (1)

• Evidence for dominance of hadronization by quark recombination from a thermal, deconfined phase comes from:

– Large baryon/meson ratios at moderately large pT;

– Compatibility of measured abundances with statistical model predictions at rather large pT;

– Collective radial flow still visible at large pT.

• -meson is an excellent test case (if not from KK ).

Parton Number Scaling of Elliptic Flow

In the recombination regime, meson and baryon v2 can be obtained from the parton v2 (using xi = 1/n):

3

2

2 2

2

2

22

2

2

3v 32v2

v

1 2 v

v3 3

v

1 6 v32

and

p pt t

B

p t

Mt t

p tp t

p p

p

p

pp p

Neglecting quadratic and cubic terms, a simple scaling law holds:

2 2 2 2an vv23

vd2

v 3M B ptt

tp tpp

pp

Originally poposed by S. Voloshin

Hadron v2 reflects quark flow !

Higher Fock states don’t spoil the fun

1 2

1 2

M C qq C qqg

B C qqq C qqqg

2

2

( )1

( ) 22

( )1

( ) 22

( , , )

( , ,

(

,

, )

( , , )

)

Ba b c a b c

Ba b c g

Ma b a b

Ma b g

a b c g

a b g

x x x

x x x x

x x x x x

x x x

x x x x x x x x

x

Conclusions (2)

• Recombination model works nicely for v2(p):

– v2(pT) curves for different hadrons collapse to universalcurve for constituent quarks;

– Saturation value of v2 for large pT is universal for quarks and agrees with expectations from anisotropic energy loss;

– Vector mesons ( , K*) permit test for influence of mass versus constituent number (but note the effects of hadronicrescattering on resonances!);

– Higher Fock space components can be accommodated.

Dihadron correlations

Data: A. Sickles et al. (PHENIX) Hadrons created by recofrom a thermal medium should not be correlated.

But jet-like correlations between hadrons persist in the momentum range (pT 4 GeV/c) where recombination is thought to dominate!

( STAR + PHENIX data)

Hadron-hadron correlations

Near-side dihadroncorrelations are largerthan in d+Au !!!

Far-side correlations disappear for central collisions.

d+Au

A. Sickles et al. (PHENIX)

Sources of correlations

• Standard fragmentation

• Fragmentation followed by recombination with medium particles

• Recombination from (incompletely) thermalized, correlated medium

• But how to explain the baryon excess?

• “Soft-hard” recombi-nation (Hwa & Yang). Requires microscopic fragmentation picture

• Requires assumptions about two-body cor-relations (Fries et al.)

How serious is this?

• Original recombination model is based on the assumption of a one-body quark density. Two-hadron correlations are determined by quark correlations, which are not included in pure thermal model.

• Two- and multi-quark correlations are a natural result of jet quenching by energy loss of fast partons.

• Incorporation of quark correlations is straightforward, but introduces new parameters: C(p1, p2).

Diparton correlations

• Parton correlations naturally translate into hadron correlations.

• Parton correlations likely to exist even in the "thermal" regime, created as the result of stopping of suprathermal partons.

A plausible explanation?

Two-point velocitycorrelations among 1-2 GeV/c hadrons

away-side same-side

T. Trainor

Dihadron formation mechanisms

(1 )

( )(1 )

A Aa

A A A

A BA

A A

dz Pg E

z z z

z PD

F

z

F

z DP

12

1eff2

( )

exp2

a A B

A B

A B

g P P E

P PD

P P

S F

T

H

eff

exp A BSSP

TS

PS

A

A

A

B

B

B

Correlations - formalism

222211114

2

2

2

123

13

6

2

23

13 2

1,

2

1,

2

1,

2

1)()(

)2(qPqPqPqPWqqqdqd

V

PdPd

dNMM

Di-meson production:

)(),,( 1 in

nn pwppW 1 ,qq i ji j

C p p Partons with pairwisecorrelations

22 2

1 2 0 1 23 3 61 2

1 1 1 11 2 4 ,

(2 ) 2 2 2 2MM

qq

dN Vw P w P C C P P

d Pd P

Meson-meson, baryon-baryon, baryon-meson correlations

9 , 6 , 4BB qq MB qq MM qqC C C C C C

First results of model studies are encouraging ?

Dihadron correlations - results

Meson triggers Baryon triggers

by 100/Npart

Fixed correlation

volume

Comparison with Data

R.J. Fries, S.A. Bass, BM, nucl-th/0407102, acct’d in PRL

Conclusions – at last!

Evidence for the formation of a deconfined phase of QCD matter at RHIC:

Hadrons are emitted in universal equilibrium abundances;

Most hadrons are produced by recombination of quarks;

Hadrons show evidence of collective flow (v0 and v2);

Flow pattern (v2) is not universal for hadrons, but universal for the (constituent) quarks.

Hadron correlations from quasithermal quark correlations.