Berndt Müller
Duke University
LBNL School on
Twenty Years of Collective Expansion
Berkeley, 19-27 May 2005

Special thanks to:
• 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)
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
…or what really came as a complete surprise…
What makes baryons different from mesons ?

p
T
(GeV/c) baryons mesons behaves like meson ?
(also -meson)

Recombination was predicted in the 1980’s – Hwa, Ochiai, … q q q q q
Fragmentation
Baryon
Meson
1 q q q p
M
Recombination
Baryon
Meson
1
2 p
Q p
B
3 p
Q
S. Voloshin
QM2002

… for a thermal source
Fragmentation wins out for a power law tail
Baryons compete with mesons

• 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)

REC
p p P
1 2
“fragmentation”
“recombination”
DER
“freeze-out”

q q
QGP p
T m
M
Transition time from QGP into vacuum (in rest frame of produced hadron) is: f d / m d p
T
Allows to ignore complex dynamics in hadronization region; corrections O(m/p
T
) 2 d
Not gradual coalescence from dilute system !!!

Tutorial: Non-Relativistic Recombination
Consider system of quarks and antiquarks (no gluons!) of volume V and phase-space distribution w a
(p) = p,a p,a .
Quark-antiquark state vs. meson state: x Q x
, p p
M , P
2
V
1 e
(
1 x
1 p
2 x
2
)
V
1 e iP R
M with
R 1
2
( x x
1 2
) r x x
1 2
)
Probability for finding a meson with P and q = (p
1
-p
2
)/2 :
,
1 2
,
2
(2 ) 3
V
2
3 ( P p p
1 2
) ˆ ( )
M q
2
Number of produced mesons:
2
N V
M
(2 )
3
, , ab
M
3 1
3
3
2
3 w p w p a
(
1
) b
(
1
) ,
1 2
,
2

The meson spectrum is given by: dN
M ab
C ab
M
V
(2 ) 3 (2 ) 3 w 1 P q w 1 P q a 2 b 2
ˆ ( )
M q
2
Consider case P q, where q is of order
M
, and expand (for w a
= w b
): w 1 P q w 1 P q w 1 P
2 2 2
2 ij q q w 1
2
P
Using only lowest order term: i j w 1 P
2 i w 1
2
P j w 1
2
P dN
M
V
C
M
(2 ) 3 w 1 P
2
2
(2 ) 3
ˆ ( )
M
2 q C
M
V
(2 ) 3 w 1 P
2
2
Corrections are of order
M
2 w/Pw =
M
2 /PT for thermal quarks.

The same for baryons:
,
1 2 3
,
2
(2 ) 3
V 2
3 ( P p p p
1 2 3
) ˆ ( , )
B q s
2 where q, s are conjugate to internal coordinates r 1
2
( x
1 x
2
) x
3 a d r ' x
1 x
2
The baryon spectrum is then: dN
B abc
C abc
M
V d q d s
(2 )
3
(2 ) (2 )
3 w a
1
3
P 1
2 q s w b
1
3
P 1
2 q s w c
1
3
P q q s
2
V
C
B
(2 )
3 w 1
3
P
3
1 O (
B
2
/ PT )

For a thermal Boltzmann distribution we get
( ) exp E p T w 1 P
2 w 1 P
3
3
2 exp 2 E 1 P T
2
/ exp 3 E 1 P T
3
/ exp exp
E
M
/ T
E / T
B and therefore:
B
( ) C
B dN ( ) C
M M
1 O (
2
/ PT )

General formulation relies on Wigner functions: r
1
1
2
,
1 2
1
2 r
2 r
1
1
2
,
1 2
1
2 r
2
1
3
2
3 e 2 2 W ab
( , ; ,
1 2 1 2
)
Meson number becomes:
N
M d P d q
3 ab e iq r d Rd r d r W ab
R 1
2
, 1
2
; 1
2
, 1
2 q
M r 1 r
2
*
M r 1 r
2
Relativistic generalization (u = time-like normal of volume):
3
P u
E

Relativistic formulation using hadron light-cone frame (P = P ):
3 d k k
0
2 dk d k k with k 1
2
0 k k and k xP
E dN
M
3 d P
E dN
B
3 d p d d
P u
(2 )
3
P u
(2 )
3
,
, , dx w R xP w R x P )
M x
2
' ( , ) ( , ' ) ( , (1 x x P )
B x x
2
For a thermal distribution, ( , ) exp( p T the hadron wavefunctions can be integrated out, eliminating the model dependence of predictions. This is true even if higher Fock space states are included!

W
M
W qq
W qqg
2
1 dx dx ( x x a b a b
0
1
0 dx dx ( x x a b a b
0
1
0
1 dx dx b
( x a x b
1) ( dx dx dx ( x x x a b c a b c x x Q
1)
2
(
2
) q x q x x x x Q a b c
2 b
)
) q x q x g x a b c
1)
1)
1
1
(
( , a b
, a b
)
)
2
2
( a q a
) (
) ( b
) q a
) e
( a
) ( )
0 b
1 dx a 1
( x a
, 1 x a
)
2 e a
/ e b
/ e
( x a x b
) /
For thermal medium
1
0 dx dx dx ( x x x a b c a b c e
1)
2
(
0
1 dx dx b 2
( a
, b
,1 x a x b
)
2
, , a b c
)
2
( q a
) q
( a
) ( x g c
) e a
/ e b
/ e c
/ e
( x a x b x c
) /
W qq
W qqg e
0
1 dx a 1
( x a
,1 x b
)
2
0
1 dx dx b 2
( a
, b
,1 x a x b
)
2 e

Statistical model vs. recombination
I n the stat. model, the hadron distribution at freeze-out is given by: f P u i
E d
3
P i
2 g i
3 exp
P d
P u B
B i s with
S i I
I / T 1 i
1
For p t
, 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 p t
BUT: Elliptic flow pattern is approximately additive in valence quarks, reflecting partonic, rather than hadronic origin of flow.

Recombination vs. Fragmentation
Recombination:
Fragmentation:
E dN
M
3 d P
E dN h
3 d P d
P u
(2 )
3
, dx w R xP w R x P )
M x
2 d
1
P u dz
(2 ) 3
0 z 3
( , 1 z
) h z
Recombination… ( , ) ( , (1 ) ) exp
( , ) ( , ' ) ( , (1 x ') ) exp
P v / T
P v / T
Meson
Baryon
… always wins over fragmentation for an exponential spectrum ( z<1 ): e xp ( P v / T ) e x p( P v / zT )
… but loses at large p
T
, where the spectrum is a power law ~ (p
T
) -b

Model fit to hadron spectrum
R.J. Fries, BM, C. Nonaka, S.A. Bass (PRL )
T eff
= 350 MeV blue-shifted temperature pQCD spectrum shifted by 2.2 GeV
Corresponds to
= 0.6 !!!
Recall:
G
Q
0.5 ln(…)
0.25 ln(…)

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

Hadron Spectra II

Hadron dependence of high-p t suppression
• R+F model describes different R
AA behavior of protons and pions
• Jet-quenching becomes universal in the fragmentation region

R.J. Fries r
= 0,65 r
= 0.85
r
= 0.75
includes parton energy loss

• Evidence for dominance of hadronization by quark recombination from a thermal, deconfined phase comes from:
– Large baryon/meson ratios at moderately large p
T
;
– Compatibility of measured abundances with statistical model predictions at rather large p
T
;
– Collective radial flow still visible at large p
T
.
• -meson is an excellent test case (if not from KK ).

Parton Number Scaling of Elliptic Flow
I n the recombination regime, meson and baryon v
2 from the parton v
2
(using x i
= 1/ n): can be obtained v
M
2 p t
2v p
2
1 2 v p
2 p
2 t p t
2
2 and v
B
2 p t
3v
2 p p t
3
1 6 v p
2
3 v
2 p t
3 p p t
3
2
3
Neglecting quadratic and cubic terms, a simple scaling law holds: v
M
2 p t
2 v
2 p p t
2 an d
Originally poposed by S. Voloshin v
B
2 p t
3 v
2 p p t
3

2

1
1
( M )
( a
, b
) x x b
2
( M )
( a
, b
, g
) x x b x
2 g
2
( x x x c
) x x x c
( a
, b
, c
, g
) x x x x
2 g
M C qq C qqg
B C qqq C qqqg
2
2

• Recombination model works nicely for v
2
(p):
– v
2
(p
T
) curves for different hadrons collapse to universal curve for constituent quarks;
– Saturation value of v
2 for large p
T 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 hadronic
rescattering on resonances!);
– Higher Fock space components can be accommodated.

Data: A. Sickles et al. (PHENIX)
Hadrons created by reco from a thermal medium should not be correlated.
But jet-like correlations between hadrons persist in the momentum range
(p
T
4 GeV/c) where recombination is thought to dominate!
( STAR + PHENIX data)

A. Sickles et al. (PHENIX)
Near-side dihadron correlations are larger than in d+Au !!!
d+Au
Far-side correlations disappear for central collisions.

• Standard fragmentation
• Fragmentation followed by recombination with medium particles
• Recombination from
(incompletely) thermalized, correlated medium
• But how to explain the baryon excess?
• “Soft-hard” recombination (Hwa & Yang).
Requires microscopic fragmentation picture
• Requires assumptions about two-body correlations (Fries et al.)

• 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(p
1
, p
2
).

A plausible explanation?
• 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.
T. Trainor
Two-point velocity correlations among
1-2 GeV/c hadrons away
-side
sa me-s ide

S H
F F
F dz
A z
A
(1 z
A
) g a
P
A z
A
D ( z
A
) D z P
A B
(1 z ) P
A A
E
( a A
1
2
P
B
E )
D
P
A
P 1 P
A 2 B exp
P
B
2 T eff B
A
B
A
B
A
SS S S exp
P
A
T eff
P
B

Di-meson production: dN
MM d
3
P
1 d
3
P
2
(
V
2
2 )
6 d
3 q
1 d
3 q
2
( q
1
)
2
( q
2
)
2
W
4
1
2
P
1 q
1
,
1
2
P
1 q
1
,
1
2
P
2 q
2
,
1
2
P
2 q
2
W n
( p
1
, dN
MM
3
1 2
, p n
) w ( p i
) 1 n
V
2
(2 )
6 w
2
1
P w
1
2
1
P
2
2 2 i j
C , qq i
1 2 C C
0
4 qq j
Partons with pairwise correlations
1 1
P ,
1
2 2
P
2
Meson-meson, baryon-baryon, baryon-meson correlations
C
BB
9 C C qq
,
MB
6 C qq
, C
MM
4 C qq
First results of model studies are encouraging ?

Meson triggers Baryon triggers by 100/N part
Fixed correlation volume

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

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 (v
0
and v
2
);
Flow pattern (v
2
) is not universal for hadrons, but
universal for the (constituent) quarks.
Hadron correlations from quasithermal quark correlations.